Scientific Investigations Report 2007–5093

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

U.S. Department of the Interior

U.S. Geological Survey

DOE/ID-22202

Property-Transfer Modeling to Estimate Unsaturated 

Hydraulic Conductivity of Deep Sediments at the 

Idaho National Laboratory, Idaho
Cover: Graph showing example of measured and estimated hydraulic conductivity [K(q)] curves. 

(Photographs of core samples used to develop property transfer models courtesy of Kim S. Perkins, 

U.S. Geological Survey.)
Property-Transfer Modeling to Estimate 

Unsaturated Hydraulic Conductivity of 

Deep Sediments at the Idaho National 

Laboratory, Idaho

By Kim S. Perkins and Kari A. Winfield

Prepared in cooperation with the 

U.S. Department of Energy

Scientific Investigations Report 2007–5093

U.S. Department of the Interior

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

DIRK KEMPTHORNE, Secretary

U.S. Geological Survey

Mark D. Myers, Director

U.S. Geological Survey, Reston, Virginia: 2007

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Although this report is in the public domain, permission must be secured from the individual copyright owners to 

reproduce any copyrighted materials contained within this report.

Suggested citation:

Perkins, K.S., and Winfield, K.A., 2007, Property-transfer modeling to estimate unsaturated hydraulic conductivity 

of deep sediments at the Idaho National Laboratory, Idaho: U.S. Geological Survey Scientific Investigations 

Report 2007-5093, 22 p.
iii

Contents

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

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

Site Description.....................................................................................................................................3

Previous Investigations........................................................................................................................3

Purpose and Scope ..............................................................................................................................6

Hydraulic Conductivity Estimation Approach ...........................................................................................6

Data Sets................................................................................................................................................8

Water-Retention Parameters and Hydraulic Conductivity Estimation.........................................9

Results and Discussion...............................................................................................................................11

Root-Mean-Square Error Analysis ..................................................................................................11

Parameter Testing With Numerical Simulation ............................................................................15

Summary and Conclusions.........................................................................................................................20

Acknowledgments.......................................................................................................................................20

References Cited..........................................................................................................................................20

Figures

Figure 1. Map showing locations of the Idaho National Laboratory, Idaho, and selected 

facilities....................................................................................... 2

Figure 2. Maps showing locations of boreholes in the Radioactive Waste Management 

Complex, Idaho National Laboratory, Idaho, from which sedimentary interbed 

core samples were used to develop property-transfer models........................ 4

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

Idaho National Laboratory, Idaho, from which sedimentary interbed core 

samples were used to develop property-transfer models.............................. 5

Figure 4. Diagram showing steps used to estimate unsaturated hydraulic conductivity ...... 7

Figure 5. Diagram showing textural classification of core samples (based on the 

system of the U.S. Department of Agriculture [Soil Survey Staff, 1975]) 

used to evaluate property-transfer model for estimating unsaturated 

hydraulic conductivity [K(θ)] ............................................................... 8

Figure 6. Graph showing example of water-retention [θ(ψ)] curve showing 

components of the curve-fit model developed by Rossi and Nimmo (1994) ......... 9

Figure 7. Graphs showing unsaturated hydraulic conductivity curves estimated 

using the RNJ-PTM-Ksat model (property-transfer model-estimated saturated 

hydraulic conductivity and Rossi-Nimmo junction model parameters) ............... 12

Figure 8. Graphs showing unsaturated hydraulic conductivity curves estimated 

using the RNJ-PTM model (laboratory-measured saturated hydraulic 

conductivity and property-transfer model-estimated Rossi-Nimmo 

junction model parameters) ............................................................... 12

Figure 9. Graphs showing unsaturated hydraulic conductivity curves estimated 

using the RNJ-CF model (laboratory-measured saturated hydraulic 

conductivity and Rossi-Nimmo junction model curve fits to measured water 

retention) .................................................................................... 13
iv

Figures—Continued

Figure 10. Graphs showing unsaturated hydraulic conductivity curves estimated using the 

vG-CF model (laboratory-measured saturated hydraulic conductivity and van 

Genuchten model curve fits to measured water retention) ........................... 13

Figure 11. Graph showing comparison of root-mean-square errors in estimated hydraulic 

conductivity using RNJ-PTM-Ksat model (property transfer model-estimated 

saturated hydraulic conductivity and Rossi-Nimmo junction model parameters); 

RNJ-PTM model (laboratory-measured saturated hydraulic conductivity and 

van Genuchten model curve fits to measured water retention) ..................... 14

Figure 12. Graph showing comparison between unsaturated hydraulic conductivity 

estimates from property-transfer models in terms of root-mean-square 

error and saturated hydraulic conductivity in terms of difference between 

measured and estimated values ......................................................... 14

Figure 13. Graphs showing water retention and hydraulic conductivity curves for core 

sample ICPP-SCI-V-215 (59.70 meter) illustrating the effect of few, clustered 

water-retention data points on the estimation of hydraulic conductivity ............ 15

Figure 14. Graph showing comparison between goodness-of-fit of estimated water 

retention and goodness-of-fit of estimated unsaturated hydraulic conductivity 

in terms of root-mean-square error ...................................................... 15

Figure 15. Graphs showing water-retention and hydraulic-conductivity curves used in 

numerical simulations ..................................................................... 17

Figure 16. Diagram showing results from VS2DT model numerical simulations for one-layer 

(left) and two-layer (right) systems using RNJ-PTM-Ksat model (property transfer 

model-estimated saturated hydraulic conductivity and Rossi-Nimmo junction 

model parameters); RNJ-PTM model (laboratory-measured saturated hydraulic

conductivity and property transfer model-estimated Rossi-Nimmo junction

model parameters); RNJ-CF model (laboratory-measured saturated hydraulic

conductivity and Rossi-Nimmo junction model curve fits to measured water

retention); and vG-CF model (laboratory-measured saturated hydraulic

conductivity and van Genuchten model curve fits to measured water

retention) .................................................................................... 19

Tables

Table 1. Model variations used to estimate unsaturated hydraulic conductivity ............ 7

Table 2. Core samples used to estimate unsaturated hydraulic conductivity in 

sedimentary interbeds underlying the Idaho National Laboratory, Idaho ............ 8

Table 3. Ranges in number of measured unsaturated hydraulic conductivity points and 

average number of measured points for samples from three data sets used 

in the root-mean-square error analysis................................................... 11

Table 4. Parameter values used in numerical simulations for the one- and 

two-layer systems ........................................................................... 16

Table 5. Root-mean-square error values for water retention and hydraulic conductivity 

used to describe sedimentary interbed material in numerical simulations for a 

one- and two-layer system ............................................................... 18
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

centimeter per day (cm/d) 0.03281 foot per day

cubic centimeter (cm3

) 0.06102 cubic inch 

kilometer (km) 0.6214 mile

meter (m) 3.281 foot

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 (NAD 83).

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

Acronyms

Acronyms Meaning

ICPP Idaho Chemical Processing Plant

INL Idaho National Laboratory

INTEC Idaho Nuclear Technology and Engineering Center

PTM property-transfer model

RMSE root-mean-square error

RNJ Rossi-Nimmo junction model

RTC Reactor Technology Complex

RWMC Radioactive Waste Management Complex

SDA Subsurface Disposal Area

SRP Snake River Plain

TAN Test Area North

TRA Test Reactor Area

USGS U.S. Geological Survey

VZRP Vadose Zone Research Park
vi

This page intentionally left blank.
Abstract

The unsaturated zone at the Idaho National Laboratory is 

complex, comprising thick basalt flow sequences interbedded 

with thinner sedimentary layers. Understanding the highly 

nonlinear relation between water content and hydraulic 

conductivity within the sedimentary interbeds is one element 

in predicting water flow and solute transport processes in 

this geologically complex environment. Measurement of 

unsaturated hydraulic conductivity of sediments is costly and 

time consuming, therefore use of models that estimate this 

property from more easily measured bulk-physical properties 

is desirable. 

A capillary bundle model was used to estimate 

unsaturated hydraulic conductivity for 40 samples from 

sedimentary interbeds using water-retention parameters and 

saturated hydraulic conductivity derived from (1) laboratory 

measurements on core samples, and (2) site-specific property 

transfer regression models developed for the sedimentary 

interbeds. Four regression models were previously developed 

using bulk-physical property measurements (bulk density, the 

median particle diameter, and the uniformity coefficient) as the 

explanatory variables. The response variables, estimated from 

linear combinations of the bulk physical properties, included 

saturated hydraulic conductivity and three parameters that 

define the water-retention curve. 

The degree to which the unsaturated hydraulic 

conductivity curves estimated from property-transfermodeled water-retention parameters and saturated hydraulic 

conductivity approximated the laboratory-measured data 

was evaluated using a goodness-of-fit indicator, the rootmean-square error. Because numerical models of variably 

saturated flow and transport require parameterized hydraulic 

properties as input, simulations were run to evaluate the effect 

of the various parameters on model results. Results show 

that the property transfer models based on easily measured 

bulk properties perform nearly as well as using curve fits to 

laboratory-measured water retention for the estimation of 

unsaturated hydraulic conductivity. 

Introduction

The unsaturated zone at the Idaho National Laboratory 

(INL) consists of thick layers of fractured basalt interbedded 

with thin layers of fluvial, eolian, and lacustrine sediments. 

These sedimentary interbeds affect both vertical and horizontal 

flow through the unsaturated zone although the processes are 

not well understood. The sedimentary interbeds may affect the 

flow in several ways, including the retardation of downwardmoving water as the water encounters layer boundaries; the 

generation of perched water; the homogenization of flow 

that has been focused by basalt fractures; and the formation 

of long-range, highly conductive horizontal flow paths for 

contaminants. To better understand the flow processes, the 

role of the sedimentary interbeds needs to be determined; this 

determination typically is accomplished using numerical flow 

model simulations. Numerical flow models require input of 

two basic unsaturated hydraulic properties­—water-retention 

[θ(ψ), defined as water content (θ) as a function of matric 

pressure (ψ)] and unsaturated hydraulic conductivity [K(θ), 

defined as hydraulic conductivity (K) as a function of water 

content (θ)]. 

Because laboratory measurement of θ(ψ) and K(θ) 

requires special equipment and expertise and may require 

weeks to complete for a single core sample and because of 

the thick (as much as 200 m) unsaturated zone at the INL 

(fig. 1) and the complex stratigraphy and large site area, 

core-sample data are available for only a small fraction of 

the subsurface sediments. However, particle-size distribution 

and other bulk property data are available because, when core 

samples were not obtained during drilling, borehole cuttings 

often were collected for particle-size analysis. These particlesize distribution and other bulk physical-property data can 

be used in property-transfer models (PTMs) developed by 

Winfield (2005) to predict θ(ψ), thereby reducing the need for 

laboratory measurement of unsaturated hydraulic properties. 

The bulk physical-property data used in the PTMs includes 

bulk density (ρbulk) and the uniformity coefficient (Cu) and 

median particle diameter (d50) derived from particle-size 

analysis. 

Property-Transfer Modeling to Estimate Unsaturated 

Hydraulic Conductivity of Deep Sediments at the Idaho 

National Laboratory, Idaho

By Kim S. Perkins and Kari A. Winfield
Figure 1. Locations of the Idaho National Laboratory, Idaho, and selected facilities.

ID19_0124_INL_Fig01

BITTERROOT

RANGE

28

28

28

Mud

Lake

Birch

Camas

Creek

LEMHI RANGE

LOST RIVER RANGE

26

93

20

20

26

26

East

Middle Butte

Butte

IDAHO NATIONAL 

LABORATORY

TAN

RTC

INTEC

CFA

NRF

RWMC

PBF

22

22

Materials and

Fuels Complex

Terreton

Atomic

City

Vadose

Zone

Research

Park

Big

Lost

River

Little

Lost

River

Creek

33

33

20

93

Big Lost

River

playas

and sinks

Birch Creek

sinks

Howe

Arco

Big Southern

Butte

Experimental

Breeder

Reactor-1

CR

A1

US

26

CFA

INTEC

NRF

RWMC

TAN

RTC

PBF

EXPLANATION

IDAHO

BOISE

EASTERN

SNAKE RIVER

PLAIN IDAHO

NATIONAL

LABORATORY

Twin

Falls Pocatello

Idaho

Falls

113°00’ 45’

44°

00’

45’

43°

30’

15’ 112°30’ 112°15’

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, digital data, 1:24,000 and 1:100,000

Selected facilities at the Idaho National Laboratory

Central Facilities Area

 Idaho Nuclear Technology and Engineering Center

 Naval Reactors Facility

 Power Burst Facility

 Radioactive Waste Management Complex

 Reactor Technology Complex

 Test Area North

Boundary of Idaho National Laboratory

0

0

10 20

5 10 15 MILES

30 KILOMETERS

 Property-Transfer Modeling to Estimate Unsaturated Hydraulic Conductivity of Deep Sediments, INL, Idaho
Because θ(ψ) and K(θ) vary nonlinearly, parametric 

models commonly are used to represent the curves for those 

properties. The Rossi-Nimmo (1994) junction (RNJ) model, 

which was used in the development of the PTMs by Winfield 

(2005), requires three parameters to describe the entire range 

of ψ; therefore, three separate multiple linear-regression 

equations were developed. These multiple linear-regression 

equations were used by Winfield (2005) to develop two 

separate PTMs, one for θ(ψ) and the other for saturated 

hydraulic conductivity (Ksat) (Winfield, 2005). This study 

combines the PTMs developed by Winfield (2005) and the 

capillary-bundle model developed by Mualem (1976) to 

estimate K(θ). 

Site Description

The INL was established in 1949 under the U.S. Atomic 

Energy Commission, now the U.S. Department of Energy, for 

nuclear energy research. The INL occupies about 2,300 km2 of 

the west-central part of the eastern Snake River Plain (SRP). 

The site hosts several facilities (fig. 1) of which at least four 

have been used to generate, store, or dispose of radioactive, 

organic, and inorganic wastes. These facilities 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) Reactor Technology Complex (RTC), 

formerly known as the Test Reactor Area (TRA); and 

(4) Test Area North (TAN). 

The eastern SRP is a northeast-trending basin, 

approximately 320 km long and 80 to 110 km wide, that 

slopes gently to the southwest and is bordered by northwesttrending mountain ranges. The eastern SRP is underlain by 

interbedded volcanic and sedimentary layers that extend 

as much as 3,000 m below land surface. The sedimentary 

interbeds result from quiet intervals between volcanic 

eruptions and are of fluvial, eolian, 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, 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 basaltflow groups and 7 major sedimentary interbeds have been 

identified from boreholes drilled up to 220-m depth. Surficial 

sediments at the RWMC are composed primarily of silt and 

clay and range from 0 to 7 m thick (Rightmire and Lewis, 

1987; Anderson and Lewis, 1989). 

The climate of the eastern SRP is semiarid and the 

average annual precipitation is 22 cm. Parts of the SRP aquifer 

underlie the INL. The depth to the water table ranges from 

60 m in the northern part of the INL to about 200 m in the 

southern part (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 is primarily 

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 INL boundaries, the Big Lost River (fig. 1) is an 

intermittent stream that 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 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

Because of the site’s history of chemical-waste 

production and the potential for radionuclide migration from 

the shallow subsurface to the SRP aquifer, the role of the 

thick unsaturated zone in contaminant transport processes 

needs to be investigated. Few studies have been done in which 

hydraulic properties of deep unsaturated-zone sediments at 

the INL have been determined from core samples. Only two 

locations within the INL have been characterized, the RWMC 

(fig. 2) and the Vadose Zone Research Park (VZRP) near the 

INTEC (fig. 3). McElroy and Hubbell (1990) presented θ(ψ), 

particle size, and ρbulk data measured on sedimentary interbed 

core samples collected from eight boreholes near the RWMC. 

θ(ψ), K(θ), and bulk-physical property data from core-sample 

measurements were presented by Perkins and Nimmo (2000) 

for one borehole about 1.5 km southwest of the SDA (fig. 2). 

Measurements from an additional 16 boreholes also have 

been completed on sediment core samples collected from 

near the SDA (S. O. Magnuson, Battelle Energy Alliance, 

written commun., 2002). 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). 

Introduction 
ID19_0124_INL_Fig02

INL BOUNDARY

SPREADING AREAS

(SEE INSET A)

RADIOACTIVE WASTE

MANAGEMENT COMPLEX 

SOUTH-MON-A-009 SOUTH-SCI-V-015

UZ98-2

Big

UZ98-2 Borehole and identifier

Lost

River

USGS 118

D-10

D-15

D-02

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

EXPLANATION

Borehole and identifier

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”

EXPLANATION

Big Lost

River Sinks

Big Lost River Vadose Zone Research Park

Radioactive Waste

Management Complex

Idaho Nuclear Technology

and Engineering Center

Reactor Technology

Complex

IDAHO NATIONAL

LABORATORY

0 1 2 3 KILOMETERS

0 1 MILE

0

0

500 1,000 METERS

1,000 2,000 3,000 FEET

Figure 2. Locations of boreholes in the Radioactive Waste Management Complex, Idaho National Laboratory, 

Idaho, from which sedimentary interbed core samples were used to develop property-transfer models.

 Property-Transfer Modeling to Estimate Unsaturated Hydraulic Conductivity of Deep Sediments, INL, Idaho
ID19_0124_INL_fig03

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

Big Lost River

Big

Lost River

Big Lost

River Sinks

INSET A

Vadose Zone Research Park

Radioactive Waste

Management Complex

Idaho Nuclear Technology

and Engineering Center

Reactor Technology

Complex

IDAHO NATIONAL

LABORATORY

REACTOR TECHNOLOGY

COMPLEX

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

0

0

500 1,000 1,500 2,000 KILOMETERS

2,000 4,000 6,000 FEET

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'

2500 500 750 1,000 FEET

0 100 200 300 METERS

Base from U.S. Geological Survey, 1:24,000, vertical NGVD29, 

horizontal NAD83, Circular Butte 3 SW quadrangle, 1973

EXPLANATION

Paved roads

Dirt roads

Streams

ICPP-SCI-V-189 Borehole and identifier

Figure 3. Locations of boreholes in the Vadose Zone Research Park, Idaho National Laboratory, Idaho, from 

which sedimentary interbed core samples were used to develop property-transfer models.

Introduction 
Purpose and Scope

Development of a PTM to estimate K(θ) is important for 

regional unsaturated-flow models where data do not exist for 

every point in space. This report describes the combination 

of Mualem’s (1976) capillary-bundle model for estimating 

K(θ) and the site-specific PTMs developed by Winfield 

(2005) for predicting θ(ψ) and Ksat of the sedimentary 

interbeds underlying the INL. The Winfield (2005) PTMs 

were developed using available data for two locations within 

the INL, the RWMC (fig. 2) and the VZRP (fig. 3). Multiple 

linear-regression equations were used to estimate hydraulic 

properties from bulk-physical property data compiled from 

previous studies for 109 core samples (Winfield, 2005). 

Regression equations were developed for hydraulic parameters 

defining the θ(ψ) curve [saturated water content (θsat) and two 

parameters from the RNJ model] and Ksat. Winfield (2005) 

described the available data, data selection and processing, 

multiple linear-regression assumptions and approach, and 

regression equations developed for each parameter. The 

extension of the Winfield (2005) PTMs for estimating K(θ)

in this study includes a subset of data for 40 samples, which 

are the only samples for which laboratory-measured K(θ) data 

from minimally disturbed core samples are available (Perkins 

and Nimmo, 2000; Perkins, 2003; Winfield, 2003). 

This report presents a procedure for using the PTMs

developed by Winfield (2005) for estimating K(θ). K(θ)

curves estimated from PTM-estimated θ(ψ) and Ksat

combined with Mualem’s (1976) capillary-bundle model are 

compared to those (1) obtained with data measured directly 

in the laboratory, (2) estimated on the basis of RNJ fits to 

measured water retention, and (3) estimated on the basis of 

van Genuchten (1980) fits (referred to as vG curve fits) to 

measured water retention. Illustrative numerical simulations 

are presented to provide information on the performance of the 

PTMs in predicting unsaturated flow dynamics.

Hydraulic Conductivity Estimation 

Approach

To estimate K(θ) for the sedimentary interbeds 

underlying the INL, θ(ψ) and Ksat parameters were combined 

with Mualem’s (1976) capillary-bundle model, one of the 

most widely used K(θ) models available. Mualem’s model 

infers a pore-size distribution for a soil from its θ(ψ) curve 

based on capillary theory, which assumes that a pore radius 

is proportional to the ψ value at which that pore drains. 

Mualem’s model conceptualizes pores as pairs of capillary 

tubes whose lengths are proportional to their radii; the 

conductance of each capillary-tube pair is determined 

according to Poiseuille’s law1. In this formulation, K(θ) is 

defined as

K K K

K

r sat

r

( ) ( ) ,

( )

θ θ

θ

=

where 

 is relative hydraulic conductivity.

(1)

To compute K(θ) for the whole medium, the conductance 

of all capillary-tube pairs is integrated as

K

d

d

S

r e

L

e

sat

θ ψ θ

θ

ψ θ

θ

θ

θ

θ ( ) =

























−

∫

∫

S

where 

= 

1

1

0

0

2

( )

( )

,

θ

θ θ

θ

sat r

r

− ,

 is residual water content expressed 

volumetrically, 

 is saturated water content expressed 

volumetr

θsat

ically,

 is a dimensionless parameter interpreted as 

repr

L

esenting the tortuosity and connectivity

of pores with different sizes and usually is

given the value of 0.5, and

ψ θ( ) is the retention curve with matric pressure 

expressed as a function of water content. 

(2)

Laboratory-measured K(θ) values were compared to 

those estimated by equation 1 using four model variations: 

(1) PTM-estimated Ksat and RNJ model parameters (referred 

to as the RNJ-PTM-Ksat model), (2) laboratory-measured Ksat

and PTM-estimated RNJ model parameters (referred to as 

the RNJ-PTM model), (3) laboratory-measured Ksat and RNJ 

model curve fits to measured water retention (referred to as 

the RNJ-CF model), and (4) laboratory-measured Ksat and 

vG model curve fits to measured water retention (referred to 

as the vG-CF model). The basic steps for estimating K(θ) are 

outlined in figure 4 and table 1. 

1 Poiseuille’s law states that the flow rate per unit cross-sectional area of a 

capillary tube is proportional to the square of the radius.

 Property-Transfer Modeling to Estimate Unsaturated Hydraulic Conductivity of Deep Sediments, INL, Idaho
Figure 4. Steps used to estimate unsaturated hydraulic conductivity.

ID19_0124_INL_Fig04

UNSATURATED HYDRAULIC

CONDUCTIVITY

K(θ)

Water-retention

[θψ]

parameters

Mualem (1976) model

Saturated hydraulic

conductivity

(Ksat)

Property transfer

model-estimated

saturated hydraulic

conductivity (Ksat)

K(θ) estimates:

Input for K(θ)

estimates:

van Genuchten (1980)

model curve fits

to measured

water retention θ(ψ)

Rossi-Nimmo (1994)

junction model curve

fits to measured

water retention θ(ψ)

Source of

input:

1 2 3 4 5

Model variations

evaluated:

1 and 3 (referred to as the RNJ-CF model)

1 and 4 (referred to as the vG-CF model)

1 and 5 (referred to as the RNJ-PTM model)

2 and 5 (referred to as the RNJ-PTM-Ksat model)

Laboratory

measured

Ksat

Property transfer 

model-estimated 

Rossi-Nimmo (1994)

junction model 

parameters

Table 1. Model variations used to estimate unsaturated hydraulic conductivity.

[Abbreviations: RNJ-PTM-Ksat, property-transfer model-estimated saturated hydraulic conductivity and Rossi-Nimmo junction model parameters; RNJ-PTM, 

laboratory-measured saturated hydraulic conductivity and property-transfer model-estimated Rossi-Nimmo junction model parameters; RNJ-CF, laboratorymeasured saturated hydraulic conductivity and Rossi-Nimmo junction model curve fits to measured water retention; vG-CF, laboratory-measured saturated 

hydraulic conductivity and van Genuchten model curve fits to measured water retention]

Source of input Laboratorymeasured Ksat

Property-transfer 

model-estimated 

saturated hydraulic 

conductivity (Ksat)

Model curve fits to measured 

water retention θ(ψ)

Property-transfer 

model-estimated

Rossi-Nimmo 

junction model 

parameters

Rossi-Nimmo

junction van Genuchten 

RNJ-PTM-Ksat model X X

RNJ-PTM model X X

RNJ-CF model X X

vG-CF model X X

Hydraulic Conductivity Estimation Approach 
Data Sets

A subset of the data for the core samples used by 

Winfield (2005) to develop PTMs for θ(ψ) and Ksat was used 

to estimate K(θ). The 40 minimally disturbed core samples 

(table 2) used in this study were chosen because they are 

the only samples for which θ(ψ) and K(θ) were measured 

directly in the laboratory. The laboratory-measured θ(ψ) and 

K(θ) values for these deep core samples, in addition to bulk 

physical-property data, are presented by Perkins and Nimmo 

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

the RWMC subsurface disposal area (SDA) (fig. 2), and by 

Perkins (2003) and Winfield (2003) for seven boreholes near 

the current percolation pond area for the INTEC facility at the 

VZRP (fig. 3). The U.S. Department of Agriculture’s (USDA) 

textural classifications (Soil Survey Staff, 1975) for all core 

samples used in this study are presented in figure 5. Materials 

with high clay and gravel content known to exist within 

the sedimentary interbeds underlying the INL are underrepresented in this subset.

Figure 5. Textural classification 

of core samples (based on the 

system of the U.S. Department of 

Agriculture [Soil Survey Staff, 1975]) 

used to evaluate property-transfer 

model for estimating unsaturated 

hydraulic conductivity [K(θ)].

Table 2. Core samples used to estimate unsaturated hydraulic conductivity in sedimentary interbeds underlying the Idaho 

National Laboratory, Idaho.

[Abbreviations: RWMC, Radioactive Waste Management Complex; VZRP, Vadose Zone Research Park]

Borehole identification Borehole location Depth interval 

(meter)

Number of 

core samples Study

UZ98-2 RWMC 42.98–50.30 18 Perkins and Nimmo (2000)

ICPP-SCI-V-215 VZRP 45.54–59.92 12 Perkins (2003)

ICPP-SCI-V-189, ICPP-SCIV-198, ICPP-SCI-V-204, ICPPSCI-V-205, ICPP-SCI-V-213, 

ICPP-SCI-V-214

VZRP 36.59–56.33 10 Winfield (2003)

Total number of 

samples

40

PERCENT CLAY

PERCENT SAND

Radioactive Waste Management Complex: Perkins 

and Nimmo (2000)

Data source

Clay

Silty

clay

Silt

Silt loam

Sandy

clay

Sandy clay

loam

Sandy loam Sand

Loamy

sand

Loam

Silty clay

loam

Clay

loam

100

90

80

70

60

50

40

30

20

10

0

9080706050403020100 100

EXPLANATION

Vadose Zone Research Park: Perkins (2003)

Vadose Zone Research Park: Winfield (2003)

 Property-Transfer Modeling to Estimate Unsaturated Hydraulic Conductivity of Deep Sediments, INL, Idaho
Water-Retention Parameters and Hydraulic 

Conductivity Estimation

In the development of the PTMs by Winfield (2005), 

the RNJ model was used to fit laboratory-measured θ(ψ)

values. The RNJ model is more physically realistic for the 

entire range of θ from saturation to oven dryness (ψd) than 

other parametric models (for example, Brooks and Corey, 

1964, and van Genuchten, 1980) that include the empirical, 

optimized residual water content (θr) parameter, which is not 

well defined. According to capillary theory the largest pores, 

associated with ψ values near zero, drain first. Drainage of 

those pores is followed by drainage of successively smaller 

pores as θ approaches θr. The Brooks and Corey and van 

Genuchten models have an asymptotic approach to θr meaning 

that the number of small pores approaches infinity, which 

is physically unrealistic. The θ(ψ) curve represented by the 

RNJ model does not have a parameter analogous to θr

; the 

curve goes to zero θ at a fixed ψ value calculated for the 

conditions of ψd. The RNJ model, like many other parametric 

water-retention models, can be combined analytically with 

the capillary-bundle model developed by Mualem (1976) to 

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

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

The RNJ model consists of three functions joined at 

two points (defined as ψi and ψj, fig. 6). These functions are 

a parabolic function for the wet range of ψ (ψi ≤ ψ ≤ 0), a 

power-law function (Brooks and Corey, 1964) for the middle 

range of ψ ( ψj ≤ ψ ≤ ψi), and a logarithmic function for the 

dry range of ψ ( ψd ≤ ψ ≤ ψj). The model has two independent 

parameters: (1) the scaling factor for ψ (ψo), and (2) the 

curve-shape parameter (λ). Sometimes, ψo is associated with 

the ψ value at which air first enters a porous material during 

desaturation (referred to as air-entry pressure). In actuality, 

however, air begins displacing water in the largest pores at a 

higher (less negative) ψ than ψo, as evidenced by the departure 

of θ from saturation to the right of ψo on the θ(ψ) curve 

(fig. 6). Unlike the model of Brooks and Corey, which holds 

θ fixed between ψ = 0 and the air-entry pressure, the RNJ 

model produces a smooth curve near saturation, represented 

by a parabolic function, that allows the pore-size distribution 

[the first derivative of the θ(ψ) curve] to be represented more 

realistically. The curve-shape parameter λ indicates the relative 

steepness of the middle part of the θ(ψ) curve, described by 

the power-law function. Larger λ values cause the drainage 

part of the θ(ψ) curve to appear steeper. 

Figure 6. Example of water-retention [θ(ψ)] curve showing components of the curve-fit model developed by Rossi and Nimmo 

(1994).

EXPLANATION

1

1

2

2

3

3

θsat

θsat

ψo

ψo

λ

λ

ψi

ψi

ψj

ψj ψd

ψd

MATRIC PRESSURE (ψ)

WATER CONTENT (θ)

-107 0

0

Parabolic function, ψi≤ ψ ≤ 0

Power-law function, ψj

≤ ψ ≤ψi

Logarithmic function, ψd ≤ ψ ≤ ψj

Saturated water content

Scaling parameter for matric pressure

Curve-shape parameter

Junction point for 1 and 2

Junction point for 2 and 3

Oven-dry matric pressure

Hydraulic Conductivity Estimation Approach 
The parabolic function applies for ψi ≤ ψ ≤ 0 and is 

represented by

θ

θ

ψ

sat ψo

c

c

= − 



 



1 

2

,

where 

 is a dimensionless constant calculated from an 

analytical function involving the parameter

 (described below), which also is 

 dimensionless. 

λ

 

(3)

The power-law function applies for ψj ≤ ψ ≤ ψi and is 

represented by

θ

θ

ψ

ψ

λ

sat

o = 



 



 .

 

(4)

The logarithmic function applies for ψd≤ ψ ≤ ψ j and is 

represented by

θ

θ

α ψ

sat ψ

d = 



 



 ln .

 

(5)

The dependent parameters are calculated as follows:

α λ ψ

ψ

ψ ψ λ

ψ ψ

λ

λ

λ

λ

= 



 





= +





 





=

= +

−

e

e

c

o

d

i o

j d

,

,

,

.

2

2

0 5 2

2

1

-1

 and

λ

λ

 λ



 





+2

.

 

(6)

For convenience, a ψ d value of -1 × 10 7 cm-water (the 

pressure at which the curve goes to zero θ) was used in the 

model fits for all core samples. That value is considered 

reasonable for a soil dried in an oven at 105–110°C with 

50-percent relative humidity (Ross and others, 1991; Rossi and 

Nimmo, 1994).

Winfield (2005) formulated the following propertytransfer equations for the RNJ model parameters and Ksat

using particle-size statistics [median particle diameter (d50) 

and uniformity coefficient (Cu)] and ρbulk as input

q ρ

ψ

sat bulk u = − 1 0063 0 3998 0 0123 0 0029 . . − . log( ) . d50 − log( ), C

log( o bulk u ) . . . log( ) . d log( ) C

l

= − + − 1 4080 1 5344 0 8394 0 1510 ρ 50 − , 

and

oog( ) . λ = − + 0 0411 0 0974 0 1925 0 4910 . ρ + . log( ) . − log( ). bulk 50 u d C

(7)

The Ksat model equation is

log( ) . .

. log( ) . log( )

K

d C

sat bulk

u

= − +

+ −

1 7690 0 0794

1 7507 0 3274 50

ρ

. (8)

The RNJ model is integrable in closed form for use in 

the Mualem (1976) model (Rossi and Nimmo, 1994). In the 

Mualem model, Κr(θ), the ratio between the unsaturated and 

saturated hydraulic conductivity can be expressed as 

K I

I r

sat sat

( ) ( )

( )

θ , θ

θ

θ

θ = 2

2

(9)

where

I I

I I

I I

j

j i

θ θ θ θ θ

θ θ θ θ θ

θ

( ) = ( ) ≤ ≤

( ) = ( ) ≤ ≤

( ) =

III

II

 for ,

 for , and

I ( ) θ θ θ θ for , i ≤ ≤ sat

(10)

and

I

I I

d sat

j

o

III

II III

1

θ α

ψ α

θ

θ

θ θ

ψ

( ) = 



 



 − 















( ) = ( ) +

exp , 1 1

λ

λ

θ

θ

θ

θ

λ

λ

λ

λ

+





 



 −





 

























( ) ( ) +

1

1

sat

j

sat

I

+1

I

and

,

θ θ

ψ

θ

θ

θ

θ ( ) = ( ) + − 



 



 − −





 











 I c

i

o

i

sat sat

II

2 1 1

1 2 1 2 1 2 / / / 







= ( ) = ( )

,

θ θ ψ θ θ ψ . j j i i 

(11)

For comparison purposes, the measured θ(ψ) data also 

were fit with the empirical formula of van Genuchten (1980), 

which has the form

θ ψ θ θ θ αψ ( ) + ( ) + ( ) 

 











 = - r sat r

n m

/ 1 , (12)

where α, n, and m are empirical, dimensionless fitting 

parameters.

10 Property-Transfer Modeling to Estimate Unsaturated Hydraulic Conductivity of Deep Sediments, INL, Idaho
Using measured θ and ψ values, α and n parameters are 

optimized to achieve the best fit to the data. The parameter 

m is set equal to 1-1/n to reduce the number of independent 

parameters and, thus, allow for better model convergence 

and to permit convenient mathematical combination with 

Mualem’s (1976) model (van Genuchten, 1980) as follows

K K S S sat e

L

e

n n n ( ) θ = − − { − 1 1 1 1 1 2 [ ] . /( - ) ( / )} (13)

Most widely used unsaturated flow and transport models 

use the van Genuchten (1980) model rather than the RNJ 

model to represent θ(ψ). The van Genuchten (1980) equation 

is parameterized by θsat, θr, α, and n, where the scaling 

parameter for ψ is α (analogous to ψo) and the curve-shape 

parameter is n (analogous to λ).

Results and Discussion

The results from the four model variations presented 

in this report were evaluated in two ways to assess model 

performance. Root-mean-square error values were calculated 

as described later to directly quantify the difference between 

laboratory-measured and model-estimated K(θ) data points. 

Because unsaturated hydraulic property parameters are 

necessary input in predictive modeling of variably saturated 

flow and transport, a numerical computer model for simulating 

unsaturated ground-water flow was used to illustrate the effect 

of parameter variation on model results.

Root-Mean-Square Error Analysis

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

as the standard error of the estimate, is used in this study as 

a goodness-of-fit indicator between measured and estimated 

K(θ) values. The RMSE is calculated as

2

1

( ) ˆ

RMSE ,

where

is the measured value,

ˆ is the estimated value of the dependent

variable and,

is the number of observations.

n

j j

j

j

j

y y

n

y 

y 

 

n 

=

−

=

∑ (14)

Small RMSE values indicate the estimated value is closer 

to the measured value of the variable. K(θ) values span several 

orders of magnitude, thus, in effect, unequally weighting points 

in the RMSE calculation. Therefore, the K(θ) values were 

transformed logarithmically prior to calculation. The number 

of K(θ) points (n) measured for each sample was between 3 and 

10. Ranges in the number of measured points and the average 

number of measured points for the samples from each data set 

are given in table 3. 

Table 3. Ranges in number of measured unsaturated hydraulic 

conductivity points and average number of measured points for 

samples from three data sets used in the root-mean-square error 

analysis.

[Abbreviations: K(θ), unsaturated hydraulic conductivity]

Study

Range in number 

of measured 

K(θ) points

Average number 

of measured 

K(θ) points

Perkins and Nimmo (2000) 3–10 6

Perkins (2003) 4–8 7

Winfield (2003) 3–8 5

Results and Discussion 11

To illustrate the performance of the four models (RNJPTM-Ksat, RNJ-PTM, RNJ-CF, and vG-CF), K(θ) curves 

are shown in figures 7–10 along with laboratory-measured 

data points for four samples from each of the three data sets 

used in this study. RMSE values were calculated for the 

40 minimally disturbed core samples using the difference 

between laboratory-measured and estimated K(θ) points to 

quantify goodness of fit. RMSE values in terms of log K(θ) 

ranged from 0.30 to 3.28 and had an average of 1.15 for the 

RNJ-PTM-Ksat model, ranged from 0.28 to 2.97 and had an 

average of 1.05 for the RNJ-PTM model, ranged from 0.21 

to 2.19 and had an average of 0.85 for the RNJ-CF model, 

and ranged from 0.35 to 9.45 and had an average of 1.61 for 

the vG-CF model. For 35 percent of the samples, K(θ) was 

best estimated by one of the property-transfer models (either 

the RNJ-PTM-Ksat or the RNJ-PTM model). For another 35 

percent, the RNJ-CF model was best, and for the remaining 

30 percent, the vG-CF model was best. For 25 of the 40 core 

samples, K(θ) was better estimated by the RNJ-CF model than 

the vG-CF model. 

The statistical distributions of RMSE values are shown 

in figure 11. The distributions indicate that the differences 

between the RNJ-PTM-Ksat and RNJ-PTM models are small 

and that using a laboratory-measured Ksat value only slightly 

improves the K(θ) estimates. The lack of correlation between 

K(θ) values estimated from PTMs in terms of the root-meansquare error and Ksat values in terms of the difference between 

the measured and estimated values is shown in figure 12. 
ID19_0124_INL_Fig07

0 0.10 0.20 0.30 0.40 0.50 0.60

VOLUMETRIC WATER CONTENT

1

HYDRAULIC CONDUCTIVITY, IN CENTIMETERS PER SECOND

Measured, ICPP-SCI-205, 42.30 m

Estimated

Measured, ICPP-SCI-V-213, 56.08 m

Estimated

Measured, ICPP-SCI-V-214, 39.40 m

Estimated

Measured, ICPP-SCI-V-214, 56.24 m

Estimated

10-10

10-8

10-6

10-4

10-2

1

10-10

10-8

10-6

10-4

10-2

1

10-10

10-8

10-6

10-4

10-2

Measured, ICPP-SCI-V-215, 45.85 m

Estimated

Measured, ICPP-SCI-V-215, 46.37 m

Estimated

Measured, ICPP-SCI-V-215, 47.42 m

Estimated

Measured, ICPP-SCI-V-215, 59.92 m

Estimated

Measured, UZ98-2, 42.98 m

Estimated

Measured, UZ98-2, 45.21 m

Estimated

Measured, UZ98-2, 49.30 m

Estimated

Measured, UZ98-2, 49.99 m

Estimated

A.

B.

C.

Figure 7. Unsaturated hydraulic conductivity curves 

estimated using the RNJ-PTM-Ksat model (propertytransfer model-estimated saturated hydraulic 

conductivity and Rossi-Nimmo junction model 

parameters). Graphs A, B, and C show four samples 

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

and Winfield (2003), respectively.

ID19_0124_INL_Fig08

0 0.10 0.20 0.30 0.40 0.50 0.60

VOLUMETRIC WATER CONTENT

HYDRAULIC CONDUCTIVITY, IN CENTIMETERS PER SECOND

Measured, ICPP-SCI-205, 42.30 m

Estimated

Measured, ICPP-SCI-V-213, 56.08 m

Estimated

Measured, ICPP-SCI-V-214, 39.40 m

Estimated

Measured, ICPP-SCI-V-214, 56.24 m

Estimated

Measured, ICPP-SCI-V-215, 45.85 m

Estimated

Measured, ICPP-SCI-V-215, 46.37 m

Estimated

Measured, ICPP-SCI-V-215, 47.42 m

Estimated

Measured, ICPP-SCI-V-215, 59.92 m

Estimated

Measured, UZ98-2, 42.98 m

Estimated

Measured, UZ98-2, 45.21 m

Estimated

Measured, UZ98-2, 49.30 m

Estimated

Measured, UZ98-2, 49.99 m

Estimated

A.

B.

C.

10-10

10-8

10-6

10-4

10-2

1

1

10-10

10-8

10-6

10-4

10-2

1

10-10

10-8

10-6

10-4

10-2

Figure 8. Unsaturated hydraulic conductivity curves 

estimated using the RNJ-PTM model (laboratorymeasured saturated hydraulic conductivity and 

property-transfer model-estimated Rossi-Nimmo 

junction model parameters). Graphs A, B, and C show 

four samples each from Perkins and Nimmo (2000), 

Perkins (2003), and Winfield (2003), respectively.

12  Property-Transfer Modeling to Estimate Unsaturated Hydraulic Conductivity of Deep Sediments, INL, Idaho
ID19_0124_INL_Fig09

0 0.10 0.20 0.30 0.40 0.50 0.60

VOLUMETRIC WATER CONTENT

HYDRAULIC CONDUCTIVITY, IN CENTIMETERS PER SECOND

Measured, ICPP-SCI-205, 42.30 m

Estimated

Measured, ICPP-SCI-V-213, 56.08 m

Estimated

Measured, ICPP-SCI-V-214, 39.40 m

Estimated

Measured, ICPP-SCI-V-214, 56.24 m

Estimated

Measured, ICPP-SCI-V-215, 45.85 m

Estimated

Measured, ICPP-SCI-V-215, 46.37 m

Estimated

Measured, ICPP-SCI-V-215, 47.42 m

Estimated

Measured, ICPP-SCI-V-215, 59.92 m

Estimated

Measured, UZ98-2, 42.98 m

Estimated

Measured, UZ98-2, 45.21 m

Estimated

Measured, UZ98-2, 49.30 m

Estimated

Measured, UZ98-2, 49.99 m

Estimated

A.

B.

C.

10-10

10-8

10-6

10-4

10-2

1

1

10-10

10-8

10-6

10-4

10-2

1

10-10

10-8

10-6

10-4

10-2

Figure 9. Unsaturated hydraulic conductivity curves 

estimated using the RNJ-CF model (laboratorymeasured saturated hydraulic conductivity and RossiNimmo junction model curve fits to measured water 

retention). Graphs A, B, and C show four samples each 

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

Winfield (2003), respectively.

ID19_0124_INL_Fig10

0 0.10 0.20 0.30 0.40 0.50 0.60

VOLUMETRIC WATER CONTENT

HYDRAULIC CONDUCTIVITY, IN CENTIMETERS PER SECOND

Measured, ICPP-SCI-205, 42.30 m

Estimated

Measured, ICPP-SCI-V-213, 56.08 m

Estimated

Measured, ICPP-SCI-V-214, 39.40 m

Estimated

Measured, ICPP-SCI-V-214, 56.24 m

Estimated

Measured, ICPP-SCI-V-215, 45.85 m

Estimated

Measured, ICPP-SCI-V-215, 46.37 m

Estimated

Measured, ICPP-SCI-V-215, 47.42 m

Estimated

Measured, ICPP-SCI-V-215, 59.92 m

Estimated

Measured, UZ98-2, 42.98 m

Estimated

Measured, UZ98-2, 45.21 m

Estimated

Measured, UZ98-2, 49.30 m

Estimated

Measured, UZ98-2, 49.99 m

Estimated

A.

B.

C.

10-10

10-8

10-6

10-4

10-2

1

1

10-10

10-8

10-6

10-4

10-2

1

10-10

10-8

10-6

10-4

10-2

Figure 10. Unsaturated hydraulic conductivity 

curves estimated using the vG-CF model (laboratorymeasured saturated hydraulic conductivity and 

van Genuchten model curve fits to measured water 

retention). Graphs A, B, and C show four samples 

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

and Winfield (2003), respectively.

Results and Discussion 13
ROOT-MEAN-SQUARE ERROR (LOG K)

2

6

7

0

1

3

4

5

8

9

10

RN-PTM-Ksat RN-PTM RN-CF vG-CF

Percentile—Percentage of analyses

 equal to or less than indicated values

EXPLANATION

Outlier

90th

10th

75th

25th

50th (median)

Figure 11. Comparison of root-mean-square 

errors in estimated hydraulic conductivity using 

RNJ-PTM-Ksat model (property transfer modelestimated saturated hydraulic conductivity and 

Rossi-Nimmo junction model parameters); RNJ-PTM 

model (laboratory-measured saturated hydraulic 

conductivity and property transfer model-estimated 

Rossi-Nimmo junction model parameters); RNJ-CF 

model (laboratory-measured saturated hydraulic 

conductivityt and Rossi-Nimmo junction model 

curve fits to measured water retention); and vG-CF 

model (laboratory-measured saturated hydraulic 

conductivity and van Genuchten model curve fits to 

measured water retention).

Figure 12. Comparison between unsaturated 

hydraulic conductivity estimates from propertytransfer models in terms of root-mean-square error 

and saturated hydraulic conductivity in terms of 

difference between measured and estimated values.

ID19_0124_INL_Fig12

0

0.5

1

1.5

2

2.5

3

3.5

0.0000001 0.000001 0.00001 0.0001 0.001 0.01

ROOT-MEAN-SQAURE ERROR IN UNSATURATED 

HYDRAULIC CONDUCTIVITY (LOG K)

DIFFERENCE BETWEEN MEASURED AND ESTIMATED

SATURATED HYDRAULIC CONDUCTIVITY

A good estimate of Ksat does not necessarily improve the 

estimation of K(θ) as might be expected because the Ksat value 

determines the position of the curve on the wet end. Although 

a well-estimated Ksat value anchors the curve near the true Ksat

value on the wet end, it has no effect on the overall shape of 

the curve. 

For several vG-CF K(θ) estimates, the RMSE values were 

unusually large (fig. 11). This occurred in cases where few 

data points were available and the data clustered within a small 

range in θ. The vG-CF model is not physically realistic for 

the entire range of θ from saturation to ψd because the model 

uses θr as an optimized parameter; θr has no physical meaning. 

For the particular cases where no θ(ψ) data are available in 

the dry range and the measured points slope steeply within 

a small range in θ, the curve asymptotically approaches θr

starting from near θsat (fig. 13A). On the resulting K(θ) curve, 

K decreases sharply with little change in θ (fig. 13B). The

θ(ψ) curve represented by the RNJ model goes to zero θ at a 

fixed value of ψ calculated for the conditions of ψd; therefore, 

even when few data points are available, the relation remains 

somewhat realistic and, in turn, allows for a better estimate of 

K(θ).

The PTMs developed by Winfield (2005) were 

evaluated in terms of the RMSE between RNJ-fit and RNJ 

PTM-estimated curves for the entire range from saturation 

to ψd rather than in terms of the RMSE between measured 

and estimated points. In evaluating K(θ) estimates for this 

study, RMSE values were calculated between measured and 

estimated K(θ) data points rather than curves because fits to 

measured K(θ) data points are often poor. There appears to 

be no correlation between the RMSE values for θ(ψ) curves 

and the RMSE values for RNJ-PTM-Ksat-estimated K(θ) data 

points (fig. 14). A good estimate of θ(ψ) does not necessarily 

improve the estimate of K(θ) because θ(ψ) and K(θ) are not 

always related as described by the mathematical formulas used 

in this study. 

14  Property-Transfer Modeling to Estimate Unsaturated Hydraulic Conductivity of Deep Sediments, INL, Idaho
Figure 13. Water retention (A) and hydraulic 

conductivity (B) curves for core sample ICPPSCI-V-215 (59.70 meter) illustrating the effect of 

few, clustered water-retention data points on the 

estimation of hydraulic conductivity.

Figure 14. Comparison between goodness-of-fit 

of estimated water retention and goodness-of-fit of 

estimated unsaturated hydraulic conductivity in terms 

of root-mean-square error.

ID19_0124_INL_Fig13

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

0 0.10 0.20 0.30 0.40 0.50 0.60

VOLUMETRIC WATER CONTENT

HYDRAULIC CONDUCTIVITY, IN CENTIMETERS PER SECOND

Measured data points

Hydraulic conductivity based on Rossi-Nimmo 

 (1994) fit to measured data points

Hydraulic conductivity based on 

 Rossi-Nimmo (1994) estimated

 parameters from 

 property-transfer models

Hydraulic conductivity based on 

 van Genuchten (1980) fit to

 measured data points

0

0.10

0.20

0.30

0.40

0.50

0.60

10 10,000 1,000 0.1110100 5 106 107

MATRIC PRESSURE, IN CENTIMETERS OF WATER 

VOLUMETRIC WATER CONTENT

Measured data points

Rossi-Nimmo (1994) junction model 

 curve fit to measured data points

Rossi-Nimmo (1994) junction model 

 parameters from property- 

 transfer models

van Genuchten (1980) curve fit to 

 measured data points

B.

A.

ID19_0124_INL_Fig14

0

0.01

0.02

0.03

0.04

0.05

0 0.5 1 1.5 2 2.5 3

ROOT-MEAN-SQUARE ERROR IN LOG HYDRAULIC CONDUCTIVITY

ROOT-MEAN-SQUARE ERROR IN WATER RETENTION (VOL/VOL)

Data from Perkins and Nimmo (2000)

Data from Perkins (2003)

Data from Winfield (2003)

Results and Discussion 15

Parameter Testing With Numerical Simulation 

Development of a PTM to estimate K(θ) is important 

for regional unsaturated-flow models where data do not 

exist for every point in space. Parameterized θ(ψ) and K(θ) 

curves, representative of the modeled media, are required 

input. For this study, numerical simulations were run to 

evaluate the effect of the input parameters on modeled results. 

Parameterized unsaturated hydraulic properties [K(θ) and 

θ(ψ)] and the USGS variably saturated two-dimensional 

transport model (VS2DT) were used to simulate flow within 

a sedimentary interbed (Lappala and others, 1983; Healy, 

1990; Hsieh and others, 1999) to evaluate the effect of the 

chosen input parameters. The model was modified to allow 

for the use of the Rossi-Nimmo water-retention parameters 

(R.W. Healy, U.S. Geological Survey, oral commun. 

2006). VS2DT solves the finite-difference approximation 

to Richards’ equation (Richards, 1931) for flow and the 

advection-dispersion equation for transport. The flow equation 

is written with total hydraulic potential as the dependent 

variable to allow straightforward treatment of both saturated 

and unsaturated conditions. Several boundary conditions 

specific to unsaturated flow, including ponded infiltration, 

specified fluxes in or out, seepage faces, evaporation, and 

plant transpiration, may be used. As input, the model requires 

saturated hydraulic conductivity, porosity, parameterized 

unsaturated hydraulic conductivity and water-retention 

functions, grid delineation, and initial hydraulic conditions. 
Simulations based on properties for two core samples 

(58.6 m and 59.2 m) from borehole ICPP-SCI-V-215 located 

at the VZRP are presented in this report. Of the eight 

simulations, four are one-layer and four are two-layer systems 

each representing a single sedimentary interbed. θ(ψ) was 

represented by functions developed by Rossi and Nimmo 

(1994) and van Genuchten (1980), in all cases using Mualem’s 

(1976) model to calculate K(θ). Parameters for each of these 

functions were obtained by property-transfer modeling or 

curve fitting, as described previously in this report. The 

unsaturated-flow properties of the one- and two-layer systems 

were characterized using RNJ-PTM-Ksat, RNJ-PTM, RNJCF, and vG-CF model parameters (table 4) for a total of eight 

simulations.

Table 4. Parameter values used in numerical simulations for the one- and two-layer systems.

[Abbreviations: RNJ-PTM-Ksat, property-transfer model-estimated saturated hydraulic conductivity and Rossi-Nimmo junction model parameters; RNJ-PTM, 

laboratory-measured saturated hydraulic conductivity and property-transfer model-estimated Rossi-Nimmo junction model parameters; RNJ-CF, laboratorymeasured saturated hydraulic conductivity and Rossi-Nimmo junction model curve fits to measured water retention; vG-CF, laboratory-measured saturated 

hydraulic conductivity and van Genuchten model curve fits to measured water retention; cm3/cm3, cubic centimeter per cubic centimeter; cm-water, centimeter 

of water; cm/d, centimeter per day]

Layer

Water-retention parameters

Saturated hydraulic 

conductivity, Ksat

(cm/d)

Saturated water 

content, θsat

(cm3/cm3)

Scaling parameter for 

matric pressure, ψo

(cm-water)

Curve-shape 

parameter, λ

RNJ-PTM-Ksat

1 0.53 12.0 0.31 33.0

2 .49 26.0 .20 6.2

RNJ-PTM

1 0.53 12.0 0.31 122.8

2 .49 26.0 .20 14.1

RNJ-CF

1 0.57 3.6 0.22 122.8

2 .53 75.4 .31 14.1

Scaling parameter for 

matric pressure, α

(1/cm-water)

Curve-shape 

parameter, n

vG-CF

1 0.57 0.28 1.22 122.8

2 .53 .01 1.38 14.1

16  Property-Transfer Modeling to Estimate Unsaturated Hydraulic Conductivity of Deep Sediments, INL, Idaho

The 1- by 1-m domain was discretized into 1- by 1-cm 

grid blocks with a boundary condition chosen to simulate 

5 days of saturated infiltration within a 10-cm section at 

the top left of the domain. Initial hydraulic conditions were 

specified as uniform pressure head. 

θ(ψ) and K(θ) curves used as input for the one- and 

two-layer systems are shown in figure 15. RMSE values for 

θ(ψ) and K(θ) used to describe the sedimentary material in the 

simulations are given in table 5. The RMSE values for θ(ψ) 

were calculated on basis of differences between laboratorymeasured θ(ψ) data points and estimated ψ values for the 

given θ to allow for direct comparison between the RNJ-PTM, 

RNJ-CF, and vG-CF models. 
ID19_0124_INL_Fig15

10-9

10-8

10-7

10-6

10-5

10-4

10-3

1

10-1

10-2

0 0.10 0.20 0.30 0.40 0.50 0.60

VOLUMETRIC WATER CONTENT

HYDRAULIC CONDUCTIVITY, IN CENTIMETERS PER SECONDS 

0

0.10

0.20

0.30

0.40

0.50

0.60

10 10,000 1,000 0.1110100 5 106 107

VOLUMETRIC WATER CONTENT

C.

A.

0 0.10 0.20 0.30 0.40 0.50 0.60

10 10,000 1,000 0.1110100 5 106 107

D.

B.

Laboratory-measured data points

Rossi-Nimmo (1994) model parameters and 

Ksat estimated from property transfer models

Rossi-Nimmo (1994) model parameters estimated from 

 property transfer models, Ksat measured

Rossi-Nimmo curve (1994) fit to measured water-retention 

 data, Ksat measured

van Genuchten curve (1980) fit to water- 

 retention data, Ksat measured

MATRIC PRESSURE, IN CENTIMETERS OF WATER 

Laboratory-measured data points

Rossi-Nimmo (1994) junction model 

 parameters from property –

 transfer models

Rossi-Nimmo (1994) junction model 

 curve fit to measured data points

van Genuchten (1980) curve fit to 

 measured data points

Figure 15. Water-retention (A is layer 1, B is layer 2) and hydraulic-conductivity (C is layer 1, D is layer 2) curves 

used in numerical simulations.

Results and Discussion 17
Table 5. Root-mean-square error values for water retention and hydraulic conductivity used to describe sedimentary interbed material 

in numerical simulations for a one- and two-layer system.

[Abbreviations: RNJ-PTM-Ksat, property-transfer model-estimated saturated hydraulic conductivity and Rossi-Nimmo model parameters; RNJ-PTM, 

laboratory-measured saturated hydraulic conductivity and property-transfer model-estimated Rossi-Nimmo model parameters; RNJ-CF, laboratory-measured 

saturated hydraulic conductivity and Rossi-Nimmo model curve fits to measured water retention; vG-CF, laboratory-measured saturated hydraulic conductivity 

and van Genuchten model curve fits to measured water retention; RMSE, root-mean-square error; K(θ), unsaturated hydraulic conductivity; θ(ψ), water 

retention]

Layer RNJ-PTM-Ksat RNJ-PTM RNJ-CF vG-CF

RMSE K(θ) in units of log K

1 0.72 0.56 1.29 2.60

2 .37 .36 .67 .57

RMSE θ(ψ) in units of volumetric water content

1 0.01 0.01 0.02 0.10

2 .01 .01 .02 .14

Simulation results for all cases are presented in figure 16. 

The images show the θ distribution at the end of the 5-day 

simulation period. The results for the RNJ-PTM-Ksat and RNJPTM model simulations are qualitatively the most similar for 

the one- and two-layer systems where the only difference in 

the input parameters is the Ksat values. The lower Ksat values 

(table 4) estimated for the RNJ-PTM-Ksat model shifts the 

K(θ) curve toward higher θ for a given K relative to the curve 

for the RNJ-PTM model (fig. 15C). The effect of this shift is 

apparent in the slightly larger and more diffuse wetted area 

for the RNJ-PTM model simulations, primarily for the layer-1 

material where the Ksat is substantially higher. For the RNJ-CF 

model simulations, there is less horizontal and more vertical 

flow within the layer-1 material than in the RNJ-PTM-Ksat and 

RNJ-PTM model simulations. The RNJ-CF model parameters 

for layer 1 include a high Ksat value combined with a low ψo

value, which tends to promote vertical flow and decrease the 

driving force (suction) for horizontal flow. The RNJ-CF model 

shifts the K(θ) curve toward higher θ for a given K in layer 1 

(fig. 15C). The simulations using the vG-CF model parameters 

result in a much smaller wetted area in both cases, probably 

18  Property-Transfer Modeling to Estimate Unsaturated Hydraulic Conductivity of Deep Sediments, INL, Idaho

because of the large decrease in K with very little change in 

θ near the wet end of the K(θ) curve (fig. 15C and D). This 

artifact, which is the main difference in the curve shapes, may 

have been the cause of an apparent reduction in computational 

efficiency. The simulations using vG-CF model parameters 

required several hours, whereas the simulations using the RNJ 

model parameters required only several minutes. 

The simulation results shown in figure 16 illustrate 

the effect of the model parameters by a comparison among 

methods. The only indication of which simulations come 

closest to reality is based on laboratory-measured hydraulic 

properties and the calculated RMSE values (table 5). The 

RNJ-PTM model estimates yielded the lowest RMSE values. 

If the RNJ-PTM model simulations are assumed to be the 

most accurate, then the RNJ-PTM-Ksat model simulations 

match most closely. However, in some cases, using the RMSE 

values in this type of evaluation can be misleading; the RMSE 

is the error calculated for the entire range of points whereas 

the simulation may be most affected by the nature of the wet 

end of the curves. 
ID19_0124_INL_Fig16

0

0.1

0.2

0.3

0.4

0.5

0 0.25 0.50 0.75 1

1

0.75

0.50

0.25

0

0 0.25 0.50 0.75 1

DEPTH, IN METERS

1

0.75

0.50

0.25

0

DEPTH, IN METERS

1

0.75

0.50

0.25

0

DEPTH, IN METERS

1

0.75

0.50

0.25

0

DEPTH, IN METERS

RNJ-PTM-Ksat

RNJ-PTM

RNJ-CF

vG-CF

RNJ-PTM-Ksat

RNJ-PTM

RNJ-CF

vG-CF

Volumetric water content scale

EXPLANATION

DISTANCE, IN METERS DISTANCE, IN METERS

Figure 16. Results from VS2DT model 

numerical simulations for one-layer (left) 

and two-layer (right) systems using RNJPTM-Ksat model (property transfer modelestimated saturated hydraulic conductivity 

and Rossi-Nimmo junction model 

parameters); RNJ-PTM model (laboratorymeasured saturated hydraulic conductivity 

and property transfer model-estimated RossiNimmo junction model parameters); RNJCF model (laboratory-measured saturated 

hydraulic conductivity and Rossi-Nimmo 

junction model curve fits to measured water 

retention); and vG-CF model (laboratorymeasured saturated hydraulic conductivity 

and van Genuchten model curve fits to 

measured water retention).

Results and Discussion 19
Summary and Conclusions

Property-transfer models (PTMs) developed for 

sedimentary interbeds at the Idaho National Laboratory 

were combined with a capillary-bundle model to estimate 

unsaturated hydraulic conductivity [K(θ)]. The site-specific 

property-transfer models were developed to estimate saturated 

hydraulic conductivity (Ksat) and the three parameters that 

define the water-retention curves [θ(ψ)] using the RossiNimmo junction (RNJ) model [saturated water content (θ sat), 

a scaling parameter for matric pressure(ψo), and a curve-shape 

parameter (λ)] from more easily-measured bulk density and 

particle-size distributions. The RNJ model was integrated with 

Mualem’s capillary-bundle model for K(θ). θ(ψ) and K(θ) are 

required as input in numerical models of variably-saturated 

flow and transport which are commonly-used tools in risk 

assessment. 

K(θ) was calculated using (1) PTM-estimated Ksat and 

PTM-estimated RNJ model parameters (RNJ-PTM-Ksat), (2) 

laboratory-measured Ksat and PTM-estimated RNJ model 

parameters (RNJ-PTM), (3) laboratory-measured Ksat and 

RNJ model curve fits to measured θ(ψ) data points (RNJ-CF), 

and (4) laboratory-measured Ksat and van Genuchten model 

curve fits to measured θ(ψ) data points (vG-CF). The rootmean-square errors between laboratory-measured and modelestimated values of K(θ) for 40 samples were compared to 

evaluate the performance of the PTMsin estimating K(θ). The 

PTMs performed well; K was equally well predicted by either 

the RNJ-PTM-Ksat or RNJ-PTM models as by the RNJ-CF 

model and better overall than the vG-CF model. The RNJPTM-Ksat and RNJ-PTM models performed almost equally 

well, supporting the value of the Winfield (2005) PTMs. 

Direct measurement of Ksat may not appreciably improve the 

estimation capability of the θ(ψ) PTMs. Another significant 

conclusion is that in 25 of the 40 cases, K was better predicted 

by the RNJ-CF model than by the vG-CF model, most likely 

because of the more physically realistic nature of the RNJ 

model. Interestingly, there is no correlation between goodness 

of θ(ψ) estimation and goodness of K(θ) estimation nor is 

there a correlation between goodness of Ksat estimation and 

goodness of K(θ)estimation. 

Because PTMs commonly are used to derive input 

for ground-water flow and transport models, numerical 

simulations were run to illustrate the influence of the 

parameters on simulated results by allowing a comparison 

among methods. Conclusions drawn from this exercise are 

useful, though limited. The only indication as to which 

simulation approximates reality most closely is based on 

knowledge of laboratory-measured hydraulic properties and 

the computed root-mean-square error values, where the RNJPTM model estimations yielded the lowest values. RNJ-PTMKsat model simulation results are qualitatively the most similar 

to the RNJ-PTM model simulation results. However, using 

root-mean-square error values in this type of evaluation can 

be misleading as the error is calculated for the entire range of 

points whereas the simulation may be most affected by the 

nature of the wet end of the curves. 

A limitation to the practical application of the 

PTMs is that the core samples used to develop them are 

texturally limited; high clay and gravelly samples are 

underrepresented. However, the property-transfer models do 

afford the opportunity to estimate hydraulic properties of the 

sedimentary interbeds over greater depths and distances than 

would otherwise be possible. 

Acknowledgments

The authors would like to thank Joseph Rousseau, 

U.S. Geological Survey, for his continued support and 

encouragement during this project.

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22  Property-Transfer Modeling to Estimate Unsaturated Hydraulic Conductivity of Deep Sediments, INL, Idaho
Manuscript approved for publication, May 15, 2007.

Prepared by the USGS Publishing Network

Bill Gibbs 

Bob Crist

Bobbie Jo Richey

Cathy Martin

Sharon Wahlstrom

For more information concerning the research in this report, contact the 

Director, Idaho Water Science Center 

230 Collins Road 

Boise, Idaho 83702-4520

http://id.water.usgs.gov
Property-Transfer Modeling to Estimate Unsaturated Hydraulic Conductivity of 

Deep Sediments at the Idaho National Laboratory, Idaho

Perkins and Winfield SIR 2007–5093