A Critical Assessment of Two Phase Flow
Characterization of Soil
by
Rosanna Tse
Bachelor of Science in Civil and Environmental Engineering
University of California, Berkeley 1994
Submitted to the Department of Civil and Environmental Engineering
in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Civil and Environmental
EDgineering
at -the
Massachusetts Institute of Technology
June 1997
1997 MassaChusetts Institute of Technology.
All rights reserved.
Author
Department of Civil and Environmental Engineering
June 1997
Certified By
14
11
f
I
V
Lynn Gelhar
Professor, Civil and Environmental Engineering
Thesis Supervisor
Accepted By
Joseph M. Sussman
Chain-nan, Departmental Committee on Graduate Studies
IF
Haiti"'i
E
OF
ARCHIV S
J UN 2 41997
LIBRARIES
A Critical Assessment of Two Phase Flow
Characterization of Soils
by
Rosarma Tse
Submitted to the Department of Civil and Evironmental Engineering on
May 27,1997 in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Civil and Environmental Engineering
Abstract
Difficulties associated with the direct measurement of unsaturated hydraulic conductivity
have prompted the development of empirical models that predict unsaturated conductivity
using transformations of the more easily measured moisture retention and saturated
hydraulic conductivity data. This thesis evaluates the predictive ability of three such
models: the Brooks Corey model, the Campbell model, and the van Genuchten model.
Seven soil types totaling 71 soil samples are analyzed.
Predictive models use measured saturated hydraulic conductivity and parameters generated
from the moisture retention curve to predict the unsaturated hydraulic conductivity. A
nonlinear least-squares optimization procedure is applied to the moisture retention data to
generate the best fit parameters. Results from the analysis indicate that these models do not.
accurately predict the unsaturated conductivity. The models do not characterize the natural
variability found in aquifers. A correlation is observed between the er-rorin prediction and
the mean grain size; deviations between the measured and predicted conductivity increase as
the texture of the material becomes coarser.
Saturated conductivity is ued as the match point for the predicted models. Researchers
have suggested that a measured unsaturated conductivity point near the region of interest
will result in a better prediction. An implicit assumption within this theory is that the slope
of the predicted conductivity curve reflects the actual slope. Analysis concludes that
predicted slope does not represent the actual slope.
The use of Leverett scaling is common in modeling applications. Capillary pressure curves
are scaled by the spatia'-Iy variable saturated hydraulic conductivity in order to obtain a
single curve representative of any point within the aquifer. Results indicate that Leverett
scaling does reflect the general trends in capillarity seen at each of the sites, but does not.
represent the variability seen among individual samples at a site.
Thesis Supervisor: Lynn Gelhar
Title: Professor
Acknowledgments
First and foremost, I would like to thank the Parsons Foundation and MIT for
funding my research. Without their support, this thesis would never have been possible.
Many idividuals have contributed to this thesis. Raziuddin Khaleel of the
Westinghouse Hanford Company, John Nimmo of the U. S. Geological Survey, Peter
Wierenga of the University of Arizona, Dan Stephens of Daniel B. Stephens Associates,
Inc., and'Don Polmann of Florida's West Coast Regional Water Supply Authority
generously supplied the data used in our analysis. Richard Hills of New Mexico State
University, Paul Hsieh of the U. S. Geological Survey, and Linda Abriola of the
University of Michigan provided invaluable advice.
To Dr. Lynn Gelhar, thank you for your patience and your willingness to share
with me your imense understanding of groundwater hydrology. Your insight and depth,
of knowledge never ceases to amaze me.
To Bruce Jacobs, thank you for all the research advice and support. It was a
pleasure being your teaching assistant.
My stay at MIT is filled with fond memories. Thanks to all my friends in the
Parsons Lab who have made MIT a pleasant place to be. Special thanks goes to Julie,
Nicole, Karen F, Karen P, Ken, Guiling, Sanjay, Rob, and Dave. A super duper thanks
goes to Julie who edited my thesis.
Pat Dixon and Cynthia Stewart have been a tremendous help to me. I would never
have survived MIT without their assistance.
Living at Ashdown has been one of the best decisions I have ever made. My
thanks goes to all my friends at Ashdown, especially my fellow 4th floor residents, who
have kept me sane. Special thanks goes to my roommate, Cathy, who has been with me all
the way. Your support and encouragement has meant a lot to me. Thanks also goes to
Rebecca, Kathy, Victor, Richard, Kate, Mary, Winnie, Susan, and Inn.
Last, but definitely not least, my thanks goes to my family. Your love, support,
and encouragement has kept me going. Thank you mom, dad, grandma, Sus, Jess, Ed,
Elaine, Frances, Glenn, Ricky, Mitchell, Uncle Bill, Dan, Derek, and Carmen.
3
Table of Contents
A b stract ...........................................................................................
2
Acknowledgments..............................................................................
3
Table of Contents ...............................................................................
4
List of Figures ..................................................................................
7
List of Tables ...................................................................................
9
Chapter
10
Introduction ......................................................................
1.1 Introduction .........................................................................
10
1.2 B ackground .........................................................................
11
1.3 Relevant Literature .................................................................
1.4 Thesis Objectives ..................................................................
1.5 Thesis Outline ......................................................................
14
16
16
Chapter 2 Unsaturated Hydraulic Conductivity Models .........................17
2.1 Relevant Notation and Definitions ...............................................
2.2 Tension Dependent Models .......................................................
2.3 Saturation Dependent Models ....................................................
2.4 Pore Connectivity Models ........................................................
2.4.1 Capillary Tube Models .................................................
17
18
21
24
24
2.4.2 Rejoined Cross Section Models ......................................
25
2.5 Coupled Moisture Retention and Unsaturated Conductivity Models ........ 26
2.5.1
2.5.2
Brooks Corey ........................................................
Campbell ................................................................
27
28
2.5.3 Russo/Gardner ..........................................................
2.5.4 van Genuchten ..........................................................
29
29
4
Chapter 3 Curve Fitting Methodology .................................................
3.1 Curve Fitting Technique ..........................................................
3.1.1 Nonlinear Optimization Function.....................................
3.1.2 Optimization Output ....................................................
3.1.3 Model Parameters ......................................................
3.1.4
3.2
3.3
Initial Parameter Estimates ............................................
32
32
33
38
40
41
3.1.5 Sensitivity of Initial Parameter Estimates............................
Model Calibration..................................................................
3.2.1 RETC ....................................................................
3.2.2 Russo/Gardner ..........................................................
44
44
46
48
Data Validation Procedure for the Brooks
52
Corey Model ...................
Chapter 4 Data Description ...............................................................
Data Selection Process ............................................................
4.2 D ata Sources ........................................................................
4.2.1 Cape Cod Data Set .....................................................
4.2.2 Hanford Data Set .......................................................
4.1
54
54
55
55
56
4.2.3 Idaho National Engineering Laboratory E,;EL) Data Set ......... 58
4.2.4 Las Cruces Data Set ....................................................
61
4.2.5 Maddock Data Set ......................................................
64
4.2.6 Plainfield Data Set ......................................................
65
4.2.7
Chapter
5.1
Sevilleta Data Set .......................................................
Analysis of Curve Fitting Results ........................................
66
Unsaturated Hydraulic Conductivity Slope Analysis ..........................
66
5. 1.1 Compalison Between the Brooks
Corey and van
Genuchten Models ......................................................
5.1.2
5.2
65
Comparison Between the Brooks
67
Corey and Campbell
M ethod ...................................................................
68
Results of the Predictive Models .................................................
5.2.1 Criteria for Fit Acceptability ...........................................
70
70
5.2.2
C ape C od ................................................................
71
5.2.3 Hanford ..................................................................
72
5.2.4
78
Idaho National Engineering Laboratory (INEL) ....................
5
5.3
5.2.5 Las Cruces ..............................................................
5.2.6 Maddock .................................................................
80
81
5.2.7 Plainfield ................................................
: ...............
5.2.8 Sevilleta ..................................................................
5.2.9 Summary ................................................................
81
81
82
Match Point Selection .............................................................
90
5.4 Leverett Scaling ....................................................................
101
Chapter 6 Conclusions ......................................................................
109
6.1 Sum m ary ............................................................................
109
6.2 Future Research ....................................................................
Ill.
R e fe re n c e s ........................................................................................
113
Appendix A Table of M oisture Retention Parameters ...........................119
Appendix B Cape Cod Curve Fits ...................... ................................
125
Appendix C Hanford Curve Fits ........................................................
134
Appendix D
INEL Curve Fits ............................................................ 157
Appendix E Las Cruces Curve Fits ....................................................
174
Appendix F M addock Curve Fits .......................................................
180
Appendix G Sevilleta Curve Fits ........................................................
195
6
List of Figures
Figure 1.1: Typical Moisture Retention Curve ................................................
12
Figure 3 1: Comparison of Moisture Retention Fits Obtained withe(yf) and
io g YI(e)...............................................................................
Figure 32: Sensitivity of Calculated Hydraulic Conductivity to Residual Moisture
Content in Uniform Sand ...........................................................
34
35
Figure 33: Typical Optimized Parameter Output Box Displayedby KaleidaGraphTm
for a G eneral Curve Fit ..............................................................
39
Figure 34: Comparison of the Moisture Retention Fit Obtained Using Different
Constraints . on es .....................................................................
42
Figure 35: Comparison of Estimated Parameter Values for Hanford Sample 21636 Using Different Initial Parameter Values ................................... 45
Figure 36: Comparison of KaleidaGraphTmand RETC Generated Curve Fits for
Hanford Sample 21636 Using Different Objective Functions ................ 47
Figure 37: KaleidaGraphTmGenerated Curve Fit for Parker Silt Loam Using the
R usso M odel .........................................................................
51
Figure 4 1: (a) Fitted Moisture Retention Curve for R;EL Data Using All Tension
Points (b) Fitted Moisture Retention Curve for INEL Data Without the
Lowest Tension Point ...............................................................
60
Figure 42: van Genuchten Fitted Moisture Retention Curve for R4EL Sample ub)
80 cm . ................................................................................
Figure 43: INEL Moisture Retention Curve Showing Deviation From General
62
Shape. ................................................................................
63
Figure 5. 1: Russo Predicted Conductivity Curve for Cape Cod Core 12a.................
Figure 52: Hanford Soil Sample 0079. This sample shows (a) the difference
73
between SSHC and Ultracentrifuge measured data, (b) measured
moisture contents greater than porosity for the ultracentrifuge
measurements, and (c) an example of a good prediction. ......................
Figure 53: An Example of Scattered Conductivity Data for the Hanford Soils...........
Figure 54: Hanford Sample 21639. This sample shows (a) an example of
75
76
conductivity data in which K is ill defined and (b) an example of an
unacceptable fit. ......................................................................
7
77
Figure 5.5: The Brooks
Figure
Figure
Figure
Figure
Figure
Corey Predicted Slope Compared to the van Genuchten
Predicted Slope......................................................................
79
56: Mean Error Versus Ks for the Brooks
Corey Model ........................ 83
57: Mean Error Versus Ks for the Campbell Model ................................
84
5.8: Mean Error Versus Ks for the van Genuchten Model .......................... 85
59: Mean Error Versus llylb for the Brooks Corey Model . ....................
87
510: Mean Error Versus llige for the Campbell Model ............................. 88
Figure 5.1 1: Mean Error Versus a for the van Genuchten Model .........................
Fiaure 512: Root Mean Square Error Versus lVb for the Brooks
89
Corey
M odel ...............................................................................
Fioure 513: Root Mean Square Error Versus llyfe for the Campbell Model .............
91
92
Figure 514: Root Mean Square Error Versus a for the van Genuchten Model......... 93
Figure 5.15: Sensitivity of Calculated Hydraulic Conductivity to Saturated
Figure
Figure
Figure
Figure
516:
517:
5.18:
519:
Moisture Content for Uniform Sand ............................................
95
Regression Slope Versus llVb for the Brooks Corey Model ............ 96
Regression Slope Versus IlVe for the Campbell Model .....................97
Regression Slope Versus a for the van Genuchten Model .................. 98
Graph of n Versus a for the van Genuchten Model . ........................
103
Figure 520 (a): Graph Showing the Influence of the van Genuchten parameter n
on the Moisture Retention Curve . ...........................................
104
Figure 520 (b) Saled Moisture Retention Curves for Various Values of n at an
Effective Saturation Match Point of 13 .....................................
106
K
Figure 52 1: Graph of a Versus 0I .........................................................
8
108
List of Tables
Table 2 1: Common Variables and Definitions Found in Unsaturated Conductivity
Literature ..............................................................................
19
Table 3 1: Summary of Parameters Allowed for Each Model ................................
40
Table 32: Initial Parameter Estimates ..........................................................
43
Table 33: Comparison of KaleidaGraphTmand RETC Generated Parameter Values
Using the Same Objective Function ................................................
4.9
Table 34: Comparison of KileidaGraphTmand Russo Parameter Values .................
50
Table 4 1: Particle Size Distribution and Bulk Density for the Hanford Samples......... 57
Table 5. 1: Comparison of Calculated Slopes Between the Brooks
Corey and van
G enuchten M odels ...................................................................
9
69
Chapter I
Introduction
1 I Introduction
Numerical modeling is increasingly used to characterize the complex nature of flow and
transport in the subsurface environment. Its use in the characterization of two phase fluid
flow through the unsaturated zone of an aquifer is of particular interest. An accurate
understanding of the flow properties in the unsaturated zone is critical in estimating
contaminant transport, determining soil infiltration, and calculating recharge to the aquifer.
At present, our capacity to create complex subsurface models far exceeds our ability to
characterize the physical system it describes. The accuracy of these models rely heavily on
the quality of the measured data provided. Basic soil properties significantly influence the
outcome of these models.
The unsaturated hydraulic conductivity plays an integral role in determining the
flow characteristics of the unsaturated zone. However, the direct measurement of this.
hydraulic property is often difficult. Typical problems associated with he measurement of
the unsaturated conductivity include high costs, tedious and time consuming measurement
techniques, the hysteretical.nature of the soil properties, logistical difficulties, the immense
10
amount of data required to accurately represent the extensive variability of the soil, and the
selection. of a measurement technique that can measure conductivity values that span several
orders of magnitude.
These difficulties n the direct measurement of unsaturated conductivity have
prompted the development of empirical models that predict unsaturated conductivity. The
empirical models calculate the relative unsaturated hydraulic conductivity from more easily
measured moisture retention and saturated hydraulic conductivity data. Figure 1.1 displays
a typical moisture retention curve. The predictive models fall under three general
categories: tension dependent models, saturation dependent models, and pore connectivity
models.
Tension dependent models rely on existing measured unsaturated conductivity
values to extrapolate te rest of the conductivity curve. Saturation dependent models
predict the unsaturated conductivity curves based solely on the saturation value, but these
models do not produce unique curves for different soils. Pore connectiviv models predict
the nsaturated conductivity from the measured saturated hydraulic conductivity and
moisture retention data. A specific conductivity curve is generated for each individual soil
type. The goal of this thesis is to assess the performance of four predictive unsaturated
conductivity models derived from the pore connectivity theory.
1. 2 Background
The pore connectivity models relate unsaturated conductivity to moisture retention through
statistical analysis of the pore size distribution. Purcell 1949) and Childs and CollisGeorge 1950) are recognized as the instigators of the pore connectivity theory. Deviation
11
10
CZ
4)
4)
Main
Drainage
Curve
6m
U)
U)
(1)
6.
31
(L
inage
a
ve
1 w
iing
Is
n
Water Content
Figure I 1: Typical Moisture Retention Curve (Fmm Stephens, 1996).
12
among the various models is attributed to differences in the interpretation of pore geometry
and the estimate of its contributions to total permeability.
Models based on Purcell's theory view the pores as a bundle of straight capillaries
having a specific radii distribution function. Using Purcell's theory, Gates and Lietz
( 1950) derive a relationship between the unsaturated conductivity, the capillary tension, and
the moisture content for the wetting phase. Fatt and Dykstra 1951) and Burdine 1953)
modify Purcell to account for tortuosity in the flow path. Burdine also derives a
relationship between conductivity and moisture content for the nonwetting phase.
The Childs and Collis-George (CCG) theory models the flow through the porous
medium as flow through varying pore sizes that are randomly connected at a rejoined
interface. Wyllie and Gardner 1958) derive a relationship between conductivity and
moisture retention by simplifying the CCG theory. They assume that the pores are actually
parallel capillary tubes traversing randomly joined thin layers of soil. Mmilem.applies his
own adjustments to the CCG theory and comes up with a different relationship.
Four closed form analytical unsaturated conductivity models evolve from these,
established pore, connectivity relationships. These models predict unsaturated conductivity
using the measured saturated conductivity and parameters estimated from the moisture
retention data. Brooks
Corey 1964) characterize the moisture retention curve as a
power law and apply the Burdine relationship to generate a closed form equation for the
unsaturated conductivity. Campbell 1974) also models the moisture retention curve as a
power law, but applies the Childs and Collis-George relationship. van Genuchten 1980)
represents the moisture retention curve with a mathematical S-shape function and applies
both the Burdine and Mualem relationships to generate two different predictive unsaturated
conductivity equations. Russo 1988) assumes that unsaturated conductivity follows the
Gardner exponential model, a tension dependent model, and backtracks an expression for
the moisture retention curve using the Mualern theory.
13
1. 3 Relevant Literature
There have been many comparisons of these predictive models with measured data.
Stephens and Rehfeldt 1985) evaluate the performance of the van Genuchten model in
predicting the unsaturated conductivity for a fine sand. Measured values of moisture
content and unsaturated conductivity gravitate towards the wet range. The authors
conclude that the van Genuchten approach sufficiently predicts the conductivity for this fine
sand within the measured moisture content regime, but specifically state that predictions
may not be accurate for the dry regions. A few measurements of conductivity and moisture
retention must be made in the dry range before an absolute conclusion can be reached
regarding the overall performance of the van Genuchten model.
Stephens and Rehfeldt also conduct a series of tests to assess the sensitivity of the
conductivity predictions to the moisture cntent parameters Or, the residual moisture
content, and Os,the saturated moisture content. Results indicate that the model is highly
sensitive to the value of Or. Predicted conductivities may differ by more than one order of
magnitude depending on the choice of er. The value of Os also influences the predicted
conductivity, but the associated error is not as large as that for Or.
Russo 1988) compares the Brooks
Corey, van Genuchten, and Russo model
with two soils, a hypothetical sandy loam and a silt loam. The van Genuchten model
provides the best fit to the measured data. Russo acknowledges that model evaluations
based on just two soils is not sufficient. In order to accurately evaluate wich model works
best, model comparisons based on many ifferent soil types must be erformed.
Keuper and Frind (199 1) scale a series of moisture of retention curves for Borden
sands using a modified Leverett scaling relationship. The Leverett concept takes the
various moisture retention curves and scales them into a single curve for all the sand
14
samples. Keuper et al. use this scaled curve in their numerical analysis to represent the
moisture retention in the aquifer at any single point. A Brooks
through the scaled data. The parameters from the Brooks
Corey curve is fitted
Corey moisture-retention.
expression are then used to generate an unsaturated conductivity curve. A conductivity
curve is estimated for both the wetting and nonwetting phase from the corresponding
Brooks
Corey unsaturated conductivity expression. Numerical simulations modeling
contaminant migration are then carried out in a spatially correlated, random conductivity
field to illustrate the influence of the fluid properties.
Yates et al. 1992) analyzes the van Genuchten model on 36 soil samples taken
from 23 different soils. The soil types range from clay to sands. A majority of the -,Oil
types are fine soils. The analysis compares the measured and predicted conductivity using
the standard van Genuchten predictive method based on a match point at saturated
conductivity, the van Genuchten method with a match point at some selecied unsaturated
conductivity value, the van Genuchten method with an extra
f, and two
simultaneous fits with measured unsaturated conductivity data. The authors fnd that the
predictive approach is the least accurate of all the methods and that it introduces a
systematic bias into the unsaturated conductivity estimates. The best fit is obtained by one
of the simultaneous fits. The use of a different match point did not significantly improve
the calculated unsaturated conductivity values. Although the predictive method performed
the worst, Yates et al. stress that this does not invalidate the method. The soils studied in
this analysis do not represent the entire range of soil types. Additional soil types must be
tested before a definitive conclusion can be made.
With the exceptions of Stephens et al. and Keuper et al., all analyses of the
predictive models focus primarily on fine textured materials, such as clays and silts, at a
relatively high moisture content > 10%). Khaleel et al. specifically looks at the
performance of the van Genuchten/Mualern relationship at low moisture contents for te
15
Hanford sands. The highly heterogeneous Hanford sands contrast nicely with the
homogeneous sands previously studied. Khaleel et al. conclude that the van Genuchten
predictive method shows noticeable deviation from the measured values, especially at low
moisture contents. The use of a different conductivity match point from the unsaturated
conductivity region results in an improvement.
1.4 Thesis Objectives
The specific objectives of this study are the following:
•
To evaluate the perfon-nanceof the Brooks
Corey, Campbell, Russo, and van
Genuchten models in predicting the unsaturated conductivity of aquifer-like
materials at low moisture contents.
•
To assess the influence of changing the match point for these models.
•
To investigate the concept of Leverett scaling.
1.5 Thesis Outline
This thesis is organized into six chapters. Chapter 2 summarizes the various types of
unsaturated conductivity models and presents a detailed description of the four models used
in our analysis. Chapter 3 concentrates on the curve fitting methodology. In Chapter 4,
the data selection process is discussed and a brief description of each data set is presented.
Chapter
analyzes the curve fitting results. Chapter 6 presents the conclusions and
suggestions for future research.
16
Chapter 2
Unsaturated Hydraulic Conductivity
Aodels
This chapter summarizes the various types of models used to calculate unsaturated
hydraulic conductivity. This review follows the format presented by Mualem 1986).
Section
discusses the variables and definitions commonly found in unsaturated
conductivity equations. Sections 2 3 and 4 present the different categories of conductivity
models, namely tension dependent models, saturation dependent models, and pore
connectivity models.
Section
discusses coupled moisture retention/unsaturated
conductivity models that predict unsaturated conductivity using only measured saturated
hydraulic conductivity and moisture retention data.
2.1 Relevant Notation and Definitions
Unsaturated conductivity models attempt to predict conductivity values through measured
soil properties and characteristics, such as moisture content or capillary tension. Table 2.1
17
lists standard variables and definitions that appear ubiquitously in the unsaturated
conductivity literature.
2.2 Tension Dependent Models
When supplied with an incomplete set of unsaturated conductivity data, capillary tension
models provide a relatively straightforward and simplistic way to estimate the unknown
unsaturated conductivity values. Capillary tension models rely on existing measured
conductivities to systematically extrapolate the rest of the conductivity curve for any known
tension.
Application of the models serves multiple purposes. Use of these models can
minimize the number of measurements required for adequate representation of actual field
conditions. They also provide closed form analytical equations used to solve unsaturated
flow problems. In addition to saving time and improving accuracy, a closed form solution
simplifies the computational procedure used in numerical simulations.
One of the earliest and most widely used unsaturated hydraulic conductivity models
is formulated by Gardner 1957):
K,(V = exp(-ayf)
(2.1)
where
a
=
an empirical soil parameter
Gardner derives this analytical equation relating unsaturated hydraulic conductivity to
capillary tension as a plausible solution to Richard's Equation
193 1) for a steady state flow
scenario. Gardner 1958) determined acceptable values of a for various soil types. These
18
Standard Variables
e
actual water content
er residual water content
es
saturated water content
V
capillary pressure
K
unsaturated hydraulic conductivity
Ks saturated hydraulic conductivity
Kr
relative hydraulic conductivity
S
saturation
Se effective saturation
Standard Definitions
K =K
K,
= O 0)
1
(01 - 01)
Table 2 1: Common Variables and Definitions Found in
Unsaturated Conductivity Literature.
19
general values may be used to obtain a rough estimate of the hydraulic conductivity when
no data are available.
This exponential model is frequently employed by many soil scientists because of
its simple form. Recent research involving the stochastic analyses of steady and transient
unsaturated flow in heterogeneous media (Yeh et al., 1985; Mantoglou and Gelhar, 1987)
have adopted the Gardner exponential model to represent the local unsaturated hydraulic
conductivity.
Although simplistic in nature, the Gardner exponential model does have certain
limitations. The model does not hold well over a wide range of values. Hence, the
equation is valid only for a limited range of values of capillary tension.
Brooks and Corey
1964) suggest an alternate analytical form relating the
conductivity to capillary tension
_n
K,
for
<
K = K,
for
>
where
Yfcr
an empirically determined critical capillary tension value
n
a soil determined empirical parameter
This power law model is extremely popular in the petroleum industry.
Obvious shortcomings of the Brooks and Corey model include its high degree of
non-linearity and its inherent discontinuity near the critical capillary tension point. This
poses a distinct problem for numerical models that simulate fluid flow in the unsaturated
zone and makes it very difficult to deal with analytically. The discontinuity in the slope can
prevent rapid convergence in the model simulations.
20
Other exponential (Rijtema, 1965) and power law (Wind, 1955) models have also
been proposed. The parameters for these formulas differ slightly from Gardner and
Brooks and Corey. These formulas have not found the same degree of wide acceptance
and usage as the two models previously discussed. Tension based models also take on
other mathematical forms.
For example, King
1964) proposes a model using the
hyperbolic cosine function.
2.3 Saturation Dependent Models
Models in this category stem from the assumption that flow through the unsaturated porous
media resembles aminar flow through a capillary tube. By assuming this capillary tube
concept, the saturation based models are able to represent the microscopic flow through the
porous media by macroscopically measured flow parameters, such as hydraulic
conductivity and average velocity. Equations, such as the Hagen-Poiseuille, that link the
microscopic level to the macroscopic level form the theoretical basis behind these models.
Saturation dependent models are convenient, but they do not produce unique conductivity
curves for different soil textures.
Averjanov 1950) views unsaturated flow as flow in parallel, uniform, cylindrical
capillary tubes. The wetting fluid forms a homogeneous film along the cylinder wall,
whereas the nonwetting fluid occupies the central portion of the tube. Based on these
limiting conditions, the following relationship between unsaturated conductivity and
saturation is derived:
Kr = Se 3.5
21
Yuster 1951) essentially solves the same equations as Averjanov, but imposes
slightly different flow conditions. Yuster assumes that the nonwetting fluid in the center of
the tube flows under the same gradient as the wetting fluid. The solution is also in a power
law form, but the exponent varies:
Kr
Se 2.0
Kozeny 1927) develops a model for saturated hydraulic conductivity assuming
flow through spherical porous media:
K
=
g
=
gravity acceleration
v
=
kinematic viscosity
9 2 03
CvA'
(2.2)
where
As =
solid surface area
C = flow configuration constant
Irmay 1954) generalizes Equation 22 to represent flow through unsaturated
porous media. Since the actual values of As and C are impossible to accurately determine,
Irmay suggests that Ks act as a substitute for As and C. This yields:
Kr =Se 3
Brooks and Corey 1964) observe that Avedanov's relationship seems to agree
over a larger variety of soils than Irmay. From the above equations, one can obviously see
that slightly different interpretations in the flow conditions leads to significant differences in
22
the equation exponents. This indicates that K values may be influenced by the flow
conditions.
Mualem 1978) makes an interesting modification-to the macroscopic concept by
choosing a slightly different approach. Once again, a general power relationship is
assumed:
Kr = Sen
where
n
=
a soil parameter
Mualem attempts to detem-ine the value of n by matching the equation to experimental data
for 50 different soils. No single optimal value for n is found. Instead the analysis
indicates that a large range of possible values for n exists. Values for n span a lower limit
of 2.5 to a fairly high value of 24.5 for fine textured soils. Instead of fixing the exponent n
as a constant, Mualem defines it as a soil water characteristic parameter.
Statistical analysis of the retention data obtained from the 50 soils exhibit a
correlation between the soil parameter, n, and the energy associated with the wilting point,
W:
w
0
y. Vd 0
where
W
the energy required to drain a unit bulk volume from saturation
to the wilting point at W
7W
15000 cm
the specific weight
Using available moisture retention data, Mualem plots n versus w and derives an
empirical linear relationship between these parameters:
23
n = 30 + 0.015 w
Mualem tests this model and finds good agreement between the measured and predicted
conductivity values.
2.4 Pore Connectivity Models
Pore connectivity based models relate unsaturated hydraulic conductivity to moisture
retention and saturated conductivity measurements through statistical analysis of the pore
size distribution. The purpose of these models is to predict the conductivity curve from
measured moisture retention and saturated conductivity data. No actual measurements of
unsaturated conductivity are necessary.
Models in this category regard the porous medium as a set of randomly distributed,
interconnected pores. The moisture retention curve is viewed as the pore radii distribution
function. Flow in the porous media is controlled by the statistical probability of the random
pores connecting. Models primarily differ in their interpretation of the pore geometry and
the estimate of its contribution to total permeability. Models either regard the pores as a
bundle of capillaries or as random connections located at a rejoined cross section.
2.4.1 Capillary Tube Models
Purcell 1949) models the pores in soil as a bundle of parallel, straight capillary tubes.
Gates and Lietz 1950) apply the Purcell theory and establish the following relationship
between conductivity, capillary tension, and moisture content:
i de
0)
fo
Kr
fo,
2
V
dO
Vi2
24
Fatt and Dykstra (195 1) modify the above equation by accounting for tortuosity in
the flow path:
de
(0)
fo ""' 2+b
0
de
fo,
V 2+b
The tortuosity factor, b, varies for different soil types.
Burdine
relationship.
1953) also modifies Gates and Lietz, but use a different tortuosity
The tortuosity correction factor selected is the square of the effective
saturation.
de
K,
fo,
0 - S2
0
2
e. d
fo,
(2.3)
V2
Burdine applies the same capillary model to derive a complimentary relationship for the
nonwetting fluid, Kmw,
de
K,... (9 = (I
S) 2
(2.4)
Vf 2
de
fo,-
V2
Experimental results show good agreement between measured and predicted values using
the Burdine relationships.
2.4.2 Rejoined Cross Section Models
Childs and Collis-George 1950) model flow through a porous medium as flow through
varying pore sizes that are randomly connected at a rejoined interface. The flow is
controlled by the smaller of the two connecting pores in the sequence. Only
25
res in direct
sequence contribute to the overall conductivity. Only a single connection exists between
the pores. Various other investigators (Millington and Quirk, 1961; Jackson et al., 1965;
and so forth) have tested this theory and have modified it.
Wyllie and Gardner 1958) simplify this theory by assuming that the porous
medium is made up of randomly joined thin layers traversed by parallel capillary tubes.
The flow is controlled by the interface between the layers.
Mualem 1976) applies the Childs and Collis-George theory and arrives at the
following relationship:
0 dO
fo,
K (0) = S
(2.5)
011dO
f
V
Mualem tests this relationship on 45 soils and concludes that the optimal value for n is 0.5.
2.5 Coupled Moisture Retention and Unsaturated
Conductivity Models
This section reviews four coupled moisture retention/unsaturated conductivity models.
These models propose a mathematical equation for the moisture retention curve, apply the
equation to a pore connectivity model, and then provide a closed form analytical solution
relating the unsaturated conductivity to moisture retention and saturated conductivity. The
advantage of these models is that they can predict the unsaturated conductivity from more
easily measured moisture retention and saturated conductivity values. This thesis will
focus on how well the Brooks
Corey, Campbell, and van Genuchten models can predict
unsaturated conductivity. The Russo model is also analyzed, but the data available for this
model are limited.
26
2.5.1 Brooks
Brooks
Corey
Corey 1964) characterize the moisture retention curve as a power law
relationship:
A
S"
Vb
for
V>V
(2.6)
where
pore size distribution index
y1b
capillary tension at the bubbling pressure
The bubbling pressure is related to the maximum pore size forming a continuous network
of flow channels within the porous medium. Brooks
Corey 1964) substitute their
moisture retention equation into Equations 23 and 24, resulting in an explicit equation
relating the moisture retention and saturated conductivity to the unsaturated conductivity.
Solutions for both the wetting and nonwetting phase are derived. The wetting phase
relationships are:
(2.7)
2+3A
K = S,)
for
ig
Vfb
2+3A
K -
(2.8)
Vb
V
The nonwetting relationships are:
2+2,
(1
Se )2
(2.9)
-Se
for
A 2-
Vb
V
2+1
'9b
- -
igb
V
27
(2.10)
2.5.2 Campbell
Campbell
1974) represents the moisture retention curve by
Vf, 0
(2.11)
0.1
where
Nfe
the air entry water potential
b
an empirically determined constant
The Campbell equation differs from Brooks
Corey. First, Campbell assumes that there
is no residual moisture content. Second, the Campbell equation is valid for values of
tension below the bubbling pressure.
Campbell applies the Childs and Collis-George model and derives a conductivity
retention relationship for the wetting phase only.
2b+3
K,
0
(2.12)
e.,
K,
Ve
+2
(2.13)
b
V
Near saturation, Equation 211 faces a sharp discontinuity. This break in the retention
curve is a result&
the gradual entry of air near the saturation region. Clapp and
Hornberger 1978) suggest a modification to the moisture retention equation to account for
this discontinuity. At the inflection point (Si, Vi ) of the moisture retention curve near
saturation, Equation 2 11 is replaced by a parabolic expression
Vf= -m(S - n)(S - 1)
for
where
28
Si S5 1
(2.14)
S =:
e
e,
M
Vi
(I
n
Vib
T
2S -
Si 1
Vb
Si
1
Msi
Equation 214 passes through the point (Si VIi and (1,O) and the derivative, d VfIdS of
both Equations 211 and 214 are equal at the inflection point.
2.5.3 Russo/Gardner
Russo 1988) assumes that the relationship between conductivity and capillary pressure
follows the Gardner exponential model (Equation 2 1). An accompanying equation for the
moisture retention curve is derived by selecting Gardner's conductivity relationship as a
closed form solution to Mualem's model (Equation 25) and backing out a relationship
between saturation and capillary tension.
Se
(e -0.5aW (i+0.5aV
2
(m+2)
(2.15)
where
a
theGardnerempir.calparameter
m
2n(nisanempiricalparameterinMualem'smodel)
2.5.4 van Genuchten
van Genuchten 1980) represents the moisture retention curve with
M
Se
(2.16)
29
where
a, n, m
=empirical
parameters
van Genuchten substitutes this equation into Mualem's model (Equation 25) and solves.
A closed form analytical solution is obtained when the constraint in =
- l1n is imposed.
Solutions for the wetting phase are derived:
M-2
K,=S,2
I-
I- SM
(2.17)
when
1 (aVy-l
[1 + ,,,)n
K,
m=l-
1
n
]-,n
(2.18)
M
+(aV)' 2
Parker et al. 1987) derive the nonwetting solution:
M-2
11111
= - S,
1
1
2
(2.19)
-SIM
van Genuchten also derives relationships between the conductivity and the moisture
retention using Burdine's theory (Equation 24).
M
K,
(S.,21
1
(2.20)
SIM
when m=l-K,
+
2
n
(2.21)
11+ (,,),j2m
Dernond and Roberts 1993) derive the nonwetting counterpart
30
M
K.. = (I
e)2
(2.22)
I - SIM
At high tension, where ayl)n >> 1, the van Genuchten/Burdine combination becomes the
Brooks
Corey model. Since we are already evaluating the Brooks & Corey model, our
analysis of the van Genuchten model will focus only on the van Genuchten/Mualern
predictive equations.
31
Chapter 3
Curve Fitting 1\4ethodology
This chapter describes the curve fitting technique applied to generate the moisture retention
parameters used in predicting the unsaturated hydraulic conductivity. The first section
discusses the use of a nonlinear optimization procedure, its key parameters, and the initial
values used for the parameters. The second section presents the model calibration process.
The third section details the procedure used to select valid data points for the Brooks
Corey model.
3.1 Curve Fitting Technique
The data analysis/graphics application program KaleidaGraphTMprovided the analytical
means to evaluate the moisture retention data and produce the optimal values for the fitting
parameters. The general curve fit function in KaleidaGraphTMallows the user to define a
general form equation and its determining parameters. The program then applies a
nonlinear least-squares optimization procedure to the data values and generates the optimal
parameter values for the user defined equation. The nonlinear least-squares optimization is
32
performed using the Levenberg-Marquardt algorithm. This program presents a powerful
and effective technique for performing data analysis.
3. 1.1 Nonlinear Optimization Function
As mentioned i Chapter 2 our assessment of two phase flow characterization of soils will
focus on four coupled moisture retention/unsaturated hydraulic conductivity models. We.
are interested in how well these models can predict unsaturated hydraulic conductivity
using only moisture retention data. The models are the Brooks
Corey/Burdine model,
the Campbell model, the Russo/Gardner model, and the van Genuchten/Mualem model
(with m =
-
n).
The analysis focuses on the low moisture content region of the moisture retention
curve. We are primarily interested in the ability of these predictive models to accurately
assess the unsaturated hydraulic conductivity of the vadose zone in an aquifer. Typical
moisture contents found in the vadose zone are in the low moisture regime of the moisture
retention curve. This regime is characterized by low unsaturated hydraulic conductivity.
Selecting the form of the defining equation in our general curve fit will affect the
optimization process. If we define the moisture retention equation in terms of O(V) and
optimize on a linear scale, we lose resolution of the moisture retention curve at low
moisture contents (Figure 3 1). The residual moisture content for sand and other coarse
soils is typically just a few percent of the saturated water content. By using a inear scale to
optimize e, the low moisture content values might simply be relegated as error. This will
result in an inaccurate value for the residual water content. Stephens et al. 1985) show
that the value of the residual moisture content has a significant influence on the shape of the
predicted hydraulic conductivity curve near the low conductivity region (Figure 32).
Thus, this form of the moisture retention equation does not meet our needs.
33
05
I
-
04
-
I
C
P
Unear MolstureContent Fit
Log Transfoffned'Tension
0
C
.2'A
I . . . 1:
Fit
-
Measured Data
0
-
1000
>1
I
100
w
U
1
0
1
I
-
.
. I . I.I
0.01
0.1
I I . . . .
1
moisture content,
Figure 3 1: Comparison of Moisture Retention Fits Obtained withe(yf) and log Ve).
34
-150
-125
2M
-100
-a
E
2
I
-75
0
zfie
9
.50
a.
-25
0
Volurnetric Water Content
- A
Kr
0
-25
.50
-75
-100
-125
-150
Pressure Head (crn of water)
Figure 32: Sensitivity of Calculated Hydraulic Conductivity to Residual
Moisture Content in Uniform Sand. (From Stephens et al., 1985)
35
Defining the moisture retention equation in terms of VO) enables the optimization
process to emphasize the high tension/low moisture content region. This meets our
primary objective of highlighting the low moisture content region. However, in doing so,
it tends to neglect the fit near the low tension/high moisture content region. On a linear
scale, the measured tension data span up to four orders of magnitude. This wide range of
values makes it difficult to accurately optimize the retention curve at low tension values.
Selection of a logarithrnically transformed scale for Ve) allows us to emphasize the
low moisture content range, yet still provide a reasonable fit in the high moisture content
region (Figure 3 ). By implementing the log transformed scale, tension values are
reduced to the same order of magnitude. The curve fit procedure minimizes the relative
error between data points on a log scale as oppose to minimizing the absolute error on a
linear scale. The error values will be more uniform over the entire range of tension and will
provide a more accurate representation of the data.
The following form of the moisture retention equation for each model is used in the
optimization procedure:
Brooks
Corey
log Y/= log
log
for
Vf> V,
(3.1)
Campbell
log V/= log V - b log
(3.2)
36
Russo
log = log 0,
(O., 0')(e -0.5aV (I 0. 5aip)) m+2
(3.3)
van Genuchten
I
logv=
(3.4)
a
The KaleidaGraphTm general curve fit employs a nonlinear least-squares
optimization procedure to minimize an objective function. For the Brooks
Corey,
Campbell, and van Genuchten models, the objective function, O(b), has the general form:
N
2
I Wi[log V - 6g VI b)]l
O(b =
(3.5)
i=1
where
N
=
the number of moisture retention data in the sample
Wi
=
weighting coefficients for a single data value
log Vi
= the log of the measured tension value
15g yf (b)
=
the log of the calculated tension value
b
=
the model parameter vector
The Russo/Gardner model objective function has the form:
N
O(b = I
2
w,[log O - 6g e (b)]J
i=1
37
(3.6)
where
N
=
the number of moisture retention data in the sample
Wi
=
weighting coefficients for a single data value
log ei
=
the log of the measured content value
16g e (b)
=
the log of the calculated water content value
b
=
the model parameter vector
The weighting coefficient is used to place more or less weight on a single data value
based on a priori information regarding the reliability of the data point. In all of our
analyses, the weighting coefficient was set to unity.
The model parameter vector, b, contains the unknown coefficients in our general
equation. In the nonlinear least-squares optimization process, the parameters are adjusted
until a local minimum value of squared error is found. A more detailed discussion of the
model parameters is presented in a later section.
All algorithms and equations used in the calculation of the general curve fit can be
found in the book Numerical Recipes in C by William H. Press, Brian P. Flannery, Saul
A. Teukolsky, William T. Vetterling, Cambridge University Press (KaleidaGraphTm
reference guide).
3.1.2 Optimization Output
Once the curve fit optimization is finished, KaleidaGraphTm generates a fitted curve through
the data points and displays an output box that contains the optimized equation parameters
(Figure 33).
In addition to the parameter values and its associated error, tb..ebox also
shows the initial estimates for the parameters, the sum of the squared errors,' 2 and the
value of either R or
The sum of the squared error is calculated using the general
form:
38
KaleidaGra:ph Parameters
= g(O.01, 1.5,. 01,32)
Value
Error
0r
0.014573
0.0017608
n
1.4781
0.046843
a
as
0.32462
0.31993
0.13338
0.013818
Chisq
R
0.35178
0.99175
NA
NA
Figure 33: Typical ptimized Parameter Output Box Displayed
by KaleidaGraphrm for a General Curve Fit.
39
2
N
X2=I
- f (xi)'
Y,
a,
i
where
Yi
the measured value
f(Xd
the calculated value
ai
the weight
In our analysis, the value of the weight is set at unity.
3.1.3 Model Parameters
The model parameters for the moisture retention curve varies from model to model. Table
3.1 lists the parameters allowed for each model.
Model
Parameters
Brooks & Corey
Or,
Campbell
Ve, b
Russo
Or, Os,
van Genuchten
b
cc
11Or, Os, n, rn a
Table 3 1: Summary of Parameters Allowed for Each Model.
(Note: The parameters m and a are not the same
between the Russo and van Genuchten models)
In the van Genuchten analysis, the restricted case where m =
- 11n is selected
because it provides a simple, closed form analytical expression for the unsaturated
hydraulic conductivity. Hence, m is no longer a parameter in our optimization process for
the van Genuchten model.
40
In some instances, the parameter Os in the Russo and van Genuchten models is set
equal to the measured soil porosity. Selecting a fixed value of Osis desirable when there is
insufficient data on the moisture retention curve to define a reasonable value. In the
situation
here we have a well defined moisture retention curve, choosing Os as an
optimized parameter for the van Genuchten model will result in a more accurate definition
of the sharp downturn in the moisture retention curve near the saturation region. In
general, KaleidaGraphTM was allowed to determine the optimal value of es whenever
possible. Figure 34 illustrates the noticeable difference in the moisture r.ention
fit
acquired for the Hanford soil sample 0072 with different es constraints.
Modifications to the parameter are necessary for ill defined moisture retention
curves. For some data sets, the values of Oror Oshad to be defined. The optimized values
for these data sets did not provide reasonable values for these parameters. Negative values
of Or and values of Os greater than unity are obtained in these cases. The following chapter
indicates the data sets that require modifications to the parameters.
3.1.4 Initial Parameter Estimates
The nonlinear optimization process requires the user to specify initial estimates of the
parameter values. These initial guesses are used to generate the optimal parameter values.
It is crucial that reasonable expected values of the parameters are specified. Initial values
are chosen from suggested values found in recent literature (Khaleel et. al, 1995 and
Russo, 1988). The initial values vary from soil to soil. Different initial values for Or and
a are chosen depending on its textural classification. Compared to finer soils such as the
silts, sands generally have smaller initial guesses. For the soils where the moisture
retention curves were ill defined near the residual water content, values of Or close to its
41
Hanfopl,,dSoil 0072
5
I
I
.
.2
. I . I
.-
.
.
I
.-
I
I
.
-1 -
-
n
4
300
Q
CF
.
I
3
2
Q
1
co
Q
w
.2
0
theta s fixed
theta s variable
9
4
0
-1
f
-2
I
.
.
.
I
0.01
,
I
-- -
I
-
I -
0.1
moistwe content,
Figure 3A Comparison of the Moisture Retention Fit Obtained
Using Different Constraintson Os.
-
42
.
.
1
Models
Brooks
Initial Parameter Estimates
Corey
Or (sand)= 0.01
Or (silt)= 0. 1
= 0. 1
W
Campbell
=
We =
b
Russo
0. 1
Or
0.01
Os
porosity
M
oc
van Genuchten
Or (sand) = 0. I
Or (silt)=
0. 1
Os
porosity
n
1.5
(X
and)=
cc (silt)=
0.00 1
. I
Table 32: Initial Parameter Estimates
43
last measured moisture content value are selected. Table 32 lists the initial values generally
used for each model.
3.1.5 Sensitivity of Initial Parameter Estimates
To test the sensitivity of the optimization process to the initial parameter estimates, the
fitting procedure is performed on the same sample for different initial values (Figure 35).
For well defined moisture retention curves, the difference is negligible. For ill defined
moisture retention curves, the resulting values for Or or Os are arguably much more
sensitive to the initial specified values. In these cases, the initial value for Oris either
visually estimated from the moisture retention graph or fixed at a value deemed reasonable
for a similar soil texture. An initial guess for Os near porosity is usually sufficient to
produce a reasonable value for Os. In a few cases, the parameter Os had to be fixed.
3.2 Model Calibration
Khaleel et al. 1995) uses the computer program RETC (Leij et al., 1991) to evaluate the
performance of the van Genuchten model for the Hanford soils. RETC is a nonlinear, least
squares curve fitting procedure that optimizes specified model parameters for nonlinear
equations with multiple parameters. Our analysis uses the program KaleidaGraphT in
place of RETC.
Russo 1988) derives the Russo/Gardner model and fits it to two distinct soils, one
hypothetical and one actual. The following sections illustrate the differences between our
curve fitting procedure and those employed by Khaleel and Russo.
44
Hanford 21636
van Genuchten Curve Fit
y = vg(.018,1.7080,.1385,.30)
y = vg(O-01 15,01,32)
Value
Error
0r
0.014573
0.0017608
n
1.4781
0.046843
Oc
0.32462
0.31993
013338
0.013817
S
Chisq
R
0r
n -
a
0
Value
Error
0.014573
0.0017608
1.4781
0.046843 -
0.32462
0.31993
0.13338
0.013818
0.35178
NA
S
0.35178
0.99175
Chisq
N
R
0.99175
-
Figure 35: Comparison of Estimated Parameter Values for Hanford.
Sample 21636 Using Different Initial Parameter Values.
45
NA
3.2.1 RETC
RETC is a versatile computer program capable of finding the optimal parameters for the van
Genuchten and Brook & Corey models under varying parameter constraints. The program
allows the model parameters to be determined by using only moisture retention data or both
measured moisture retention data and unsaturated conductivity data. RETC optimizes an
alternate form of the van Genuchten equation:
0=0
(Os L
r
11
+ ayf), IM
with the objective function
N
005)
2
M
I
IWi[O- b(b)]
2
W.W2w,[Y- i(b)]J
i=,V+l
where
Oi
the measured moisture content
Oi
the calculated moisture content
Yi
the measured unsaturated hydraulic conductivities
Yi
the calculated unsaturated hydraulic. conductivities
b
the parameter vector
N
the number of moisture retention data
M
total number of measured data (retention and conductivity)
W1, W2
weighing factor between retention and conductivity data
Wi
weighing factor for a single data point
The RETC objective function optimizes O(Vf),whereas the defined KaleidaGraphrm
objective function (Equation 35) optimizes log VO). Use of different objective functions
results in slightly different calculated parameter values. Figure 36 shows the optimized
46
Hanford 21636
I
05
I
I
I . I ' ' I
I
04
to
.Iw
r-
I
.
I . I 1
RETC Fit
KaleldaGraph Fit
.2
d
I
0
1000
q
100
M
U
00
0
10
1
0.1
0.01
moisture
1
ontent 0
RETC Parameters
KaleidaGraph Parameters
0r
1
Value
Error
0.014573
0.0017608
n
ot
1.4781
0.32461
0.046843
0.13338
es
0.31993
0.013818
Chisq
0.35178
0.99175
1
Value - 0r
a
es
Chlsq
NA
NA
R2
Error
-0.022871
0.0030328
1.7077
0.073383
0.13854
0.30735
- 00005718
0.022656
0.0067038
NA
I 0.9m
I
NA
Figure 36: Comparison of MeidaGraphTl and RETC Generated Curve Fits
for Hanford Sample 21636 Using Differ-&itObjective Functions.
47
parameters generated by the two varying functions for the Hanford sample 21636.
Parameter values for the RETC curve fit are taken from Khaleel et al. 1995).
The van Genuchten parameters generated by the two methods differ slightly. The
parameters vary because the emphasis of the governing equations differ.
The
KaleidaGraphTmanalysis focuses on the low moisture content region by optimizing with
respect to log V/. RETC optimizes 0, thus concentrating on the high moisture content
region. The difference in regional emphasis is apparent in Figure 36.
To compare the optimization ability of KaleidaGraphTmagainst RETC, the RETC
objective function is entered into KaleidaGraphTm and a fitted curve is generated for
Hanford sample 21636.
Results from this test present a form of model calibration. If
KaleidaGraphTMis comparable to RETC, the generated parameter values should be the
same. RETC parameter values are taken from Khaleel et al. 1995). As seen in Table 33,
the optimized parameter values are essentially the same.
3.2.2 Russo/Gardner
To calibrate the Russo model, data for the Parker silt loam soil is read off the graph in
Russo 1988). An optimization is then performed on the measured data using Equations,
3.3 and 36. The purpose of this analysis is to replicate Russo's parameter values for this
soil. Table 34 compares the KaleidaGraphTmgenerated results versus the Russo values
found in the literature. As seen in Figure 37, the fitted curve does not accurately represent
the measured data.
Significantly different parameter values are generated in our
optimization process. Results vary because Russo uses additional data to constrain the
parameter search. Russo's objective function is
N
2
0(b)=EJwJQ(tj)-(tjb)]J
i=1
48
+JV[0(k5)-b(k5,b)]12
Parameter
KaleidaGraphrm
RETC
Or
0.022871
0.023
n
1.7077
1.7080
a
0.13854
0.1385
es
0.30735
0.3073
Table 33: Comparison of KaleidaGraphrmand RETC Generated Parameter
Values Using the Same Objective Function.
49
Parameter
I
KaleidaGraphrm_
Russo
Or
0.16136
0.186
a (M- 1)
2.128
4.995
M
1
571.88
1
0.021
i
Table 34: Comparison of KaleidaGraphrm and Russo Parameter Values
50
Slit Loam Soil
04
- - - - -
. . . . . . . . .
1000
100
ci
.21
In
C
10
>1
I9
1
2
M
IV
U
0.1
0.01
n MI
W.
J
I
-0
.9
-0.8
-0.6
-0.7
-0.5
-0.4
log moisture content, log
Russo Parameters
Value
Error
Of
0.16136
0.010189
Q
M
ChIsq
212.8
571.88
0.0091334
0.10933
113.66
NA
R2
0.95035
NA
Figure 3.7: KaleidaGraphrm Generated Curve Fit for Parker Silt
Loam Using the Russo Model.
51
where
Q(ti
(tj,
the set of cumulative outflow measurements at specified
times ti
b)
= the numerically calculated value of the outflow
corresponding to the trial vector of parameter values, b
O(k5)
= the measured moisture content at h
15,000 cm H20
0(451b)
= the predicted moisture content at h
15,000 cm H20
Wi
= weighing factor
V
= weighing factor
The parameter values from KaleidaGraphTmmight have reproduced Russo's values if extra
constraints had been added.
3.3 Data Validation Procedure for the Brooks
Model
The Brooks
Corey
Corey moisture retention relationship was discovered by plotting log S, as a
function of log y/. The resulting graph of the log transformed variables is a straight line
with the negative slope, A. The Brooks
Corey theory holds only for values of tension,
yf, greater than or equal to the tension at the bubbling pressure,
V.
In order to obtain a
proper fit to the model, points near saturation that are below the bubbling pressure tension
are not included in the curve fitting procedure.
Since the bubbling pressure tension and the residual water content are parameters in
the optimization process a graph of log S, versus log yf is not possible. A method used to
select the relevant data points is devised. The curve firwill only select these validated data
52
points in determining the slope and bubbling pressure tension value. An assumption is
made that e, is relatively small compared to the measured values of e. A graph
of og e
versus log V/is generated to represent the graph of the log S, versus log iy. In accordance
with the original Brooks
Corey method, a straight line is drawn through the data points.
Any data point near saturation that deviates from the general linear trend is excluded from
the curve fit. This procedure is used on the Hanford and R-;EL soils.
53
Chapter 4
Data Description
A total of 71 soil samples, supplied from seven data sets, is analyzed. The first section in
this chapter details the criteria used to select the data sets. The second section briefly
describes each data source, mentions the techniques used for data measurement, and
discusses any modifications made to the curve fitting parameters.
The computer disk accompanying this thesis contains the measured values of
moisture retention and unsaturated conductivity used in the analysis. AR files are saved as
Macintosh tab delineated text ffies. Each soil type has an individual file. The soil sample
number is located in the fst column of the file.
4.1 Data Selection Process
The primary purpose of the analysis is to assess the ability of these predictive models to
accurately characterize the unsaturated conductivity in aquifer like materials. Typical
aquifer materials consist of sandy, coarse soils. The relatively dry vadose zone located just
above the saturated aquifer usually resides in the low moisture content region. Hence, data
54
sets emphasizing these properties are selected. The data consists primarily of measured
moisture retention and unsaturated conductivity values obtained from sands and sandy loam
material. The Hanford data set measures capillary tension up to extremely high values,
thus giving a detailed picture of the low moisture content region.
A secondary purpose of the data selection process is to choose data sets that indicate
some measure of consistency to the fitting process. Data sets that contain multiple samples
from the same aquifer provide the repetition needed to search for consistent trends. All data.
samples in a single data set are measured by the same techniques and by the same
individuals. Thus, differences between the soil samples cannot be attributed to differences
in the measurement techniques. This gives us a good indication of the soil variability in
aquifer soils. Soil samples from the same aquifer also represent a collection of similar soils
of varying pore size distributions and saturated conductivity values.
4.2 Data Sources
This section provides a concise summary of the site description, the experimental
procedures used to measure the data values, and the changes made to the curve fitting
parameters.
For more detailed information regarding the site characterization and
measurement process for each site, the reader is advised to refer to the referenced papers.
A thorough summary of current laboratory and in-situ field measurement techniques for
moisture retention and hydraulic conductivity can be found in Hillel 1980) and Stephens
(1996).
4.2.1 Cape Cod Data Set
Soil samples from this set are taken from a glacial aquifer located on Cape Cod,
Massachusetts. The Cape Cod unconfined aquifer is a large sand and gravel outwash plain
55
that was deposited during the retreat of the continental ice sheets from southern New
England about 12,000 years ago (LeBlanc et al., 1991). This aquifer is the primary source
of freshwater for the inhabitants of Cape Cod and all its visitors. At present, the site is
contaminated by several pollutant plumes which threaten the underlying aquifer. Extensive
tests have been conducted on Cape Cod by the U. S. Geological Survey in an effort to
characterize the site.
The upper region of the aquifer is characterized as medium to coarse sand with
some gravel. Six data samples are provided by Mace 1994). The moisture retention data
focuses on the high moisture content range. Values of
range from 0095 to 023.
Insufficient data is supplied to accurately define the entire moisture retention curve. The
sharp curves near the saturated water content and the residual water content could not be
described. Hence, the model parameters er and es had to be fixed for all the models. The,
value of Os is set at the measured porosity and the value of Or is estimated at 0.01. This
value of Oris consistent with the average values of Orcalculated from the Hanford sands.
4.2.2 Hanford Data Set
The Hanford site is situated in the ad Columbia Basin located in the southeastern region
of Washington state.
It resides on the US. Department of Energy's Hanford site,
approximately 35 km northwest of Richland, Washington.
The surface soils were
deposited during a series of catastrophic glacial floods, occurring as recent as 13,000 years
ago (Khaleel et al., 1995). These glacial deposits principally consist of sands and gravels
of miscellaneous sizes. Extensive tests have been performed at the Hanford site to measure
moisture retention and unsaturated hydraulic conductivity at very low saturation values.
Capillary tension values up to 15,000 cm are measured.
56
Sample Coarse Sa& Fine Sand
(0.2 - 2
(0.02- 02
Silt
Clay
Median
SSHC Bulk
Centrifuge
(0.002 -
(<0.002
Grain Size
Density
Bulk Density
mm),
(d5O) mm
(g/cm3)
(g/cm3)
mm),
mm),
0.02 mm),
1-1417
24
68
7
I
0.095
1.67
1.79
1-1419
90
10
0
0
0.55
1.64
1.63
2-1636
85
15
0
0
0.48
1.61
1.62
2-1637
80
20
0
0
0.33
1.60
1.65
2-1638
81
19
0
0
0.60
1.72
1.82
2-1639
93
7
0
0
0.70
1.60
1.64
2-2225
80
20
0
0
0.33
1.61
1.60
2-2226
95
5
0
0
1.00
1.68
1.67
2-2227
94
6
0
0
0.72
1.67
1.63
2-2228
98
2
0
0
0.90
1.62
1.62
2-2229
95
5
0
0
0.68
1.62
1.59
2-2230
34
52
11
3
0.10
1.71
1.74
2-2232
92
8
0
0
0.68
1.71
1.71
2-2233
92
8
0
0
0.68
1.64
1.64
2-2234
86
14
0
0
0.88
1.75
1.82
0-072
27
54
10
9
0.08
1.75
1.67
0-079
0
73
22
5
0.03
1.61
1.68
0-080
8
79
8
5
0.05
1.60
1.50
0-083
38
47
8
7
0.10
1.68
1.55
0-099
58
30
7
5
0.30
1.70
1.79
0-107
80
13
5
2
0.32
1.57
1.54
0-113
74
22
1
3
0.30
1.63
1.62
Table4l: ParticleSizeDistributionandBulk-DensityfortheHanfordSamples.
57
Data
r twenty two repacked soil samples were provided by Raziuddin Khaleel of
the Westinghouse Hanford Company. Table 41 lists the particle size distribution, the
mean grain size, and the bulk density for each sample. Moisture retention data collected for
the drainage cycle span such 'awide range of values that two measurement techniques are
required. Both the pressure cell method and the pressure plate extraction method are used.
The first method measures capillary tension values up 1000 cm. The second method
measures up to 15,000 cm. Saturated hydraulic conductivity is measured using a constant
head permeameter. The unsaturated hydraulic conductivity is measured in the laboratory
using two methods: the steady state head control method and the ultracentrifuge method.
The detailed moisture retention data provided supply a complete description of the
moisture retention curves. No parameters are fixed and KaleidaGraphTm is able to optimize.
all the model parameters.
4.2.3 Idaho National Engineering Laboratory (INEL) Data Set
The Idaho National Engineering Laboratory (INEL) is located on the semi-arid eastern
Snake,River Plain in southeastern Idaho. The lab was established in 1949 as a facility to
build, operate, and test nuclear reactors.
In the southwest comer of INEL is the
Radioactive Waste Management Complex (RWMC) which acts as a storage- area for
chemical, low level radioactive and transuranic radioactive wastes. The waste is stored in
55 gallon drums and buried in trenches excavated from the surface sediments.
The eastern Snake River Plain is a structural basin underlain by basaltic rock. The
overlying surface sediments consist predominately of flood plain and wind blown deposits..
In addition to INEL, the eastern Snake River Plain also houses' one of the world's largest
aquifers, the Snake River Plain aquifer. The water table is located 180 m beneath the
surface soil in the underlying basaltic rock formation. In its natural state, the surface
sediments consist of highly structured, aggregated soil.
58
The undisturbed soil is
characterized as a distinctly layered, extremely variable soil containing macropores and a
large degree of aggregated material.
Multiple tests were conducted to ascertain the physical and hydraulic characteristics
of the surface sediments.
A simulated waste trench was constructed to represent the
RWMC trench used for radioactive waste storage. Tests were then performed on soil
samples from the simulated waste trench and on soil samples from a nearby undisturbed
area.
Moisture retention and unsaturated conductivity data are measured at four depths.
Each depth contains four samples, two measurements from the disturbed soil (soil from the
simulated trench) and two measurements from the undisturbed soil. The soils samples are
labeled in the following format: u(a) 30 cm. This format translates into undisturbed soil
sample a taken at the depth of 30 cm. A total of 16 soil samples at 4 measured depths is
available. The moisture retention and conductivity data is provided by John Nimmo of the
US. Geological Survey. Additional information regarding physical properties can be found
in Shakofsky 1995).
The INEL soil is enerally classified as a either a sandy silt or a clayey silt. The
moisture retention data is determined using a modified pressure cell method. Saturated
hydraulic conductivity is measured using the falling head method. Unsaturated hydraulic
conductivity data is generated using the one-step outflow method.
The WEL moisture retention data emphasizes data in the high moisture content
regime. Each moisture retention data set has only one low tension measurement near the
0.01 cm region. By including this point in our curve fitting analysis, we are in essence
fixing the Os parameter at that measured water content. As seen in Figure 4 (a), ill fitting
moisture retention curves are obtained with the van Genuchten model.
One of the primary goals of this analysis is to allow the best fit possible for the
moisture retention data using the selected models. In addition, the dominant area of interest
59
Using All Tension Points
0% ,
-
0
1-
I
I
I
I
I
.
I
I
2
04
0
ci
.0In
1
I
00
2t%
0.
M
Q
00
.2
0
van GerKichten
-1
0
Measured Tension
-0
II
0.1
I
moisture content,
1
Without Lowest Tension Point
3
2
1
F
0
van Genuchten
0
-2 r
0.1
Measured Temlan
moisture content,
Figure 4. 1: (a) Fitted Moisture Retention Curve for WFL Data Using AU Tension
Points (b) Fitted Moisture Retention Curve for R4EL Data Without the Lowest Tension
Point.
60
is at the low moisture content region where the unsaturated soil resides. Thus, the low
tension point near the saturated water content is neglected in the optimization process of the
van Genuchten model. Figure 4 1(b) illustrates the improved fit obtained by deleting this
point. Once again, Os is an optimized parameter. A much better curve fit develops.
Two of the soils samples, namely u(b) 30 cm and u(b) 80 cm, produce negative
values of Or. For these two cases, a value for Oris determined by looking at the moisture
retention curve and estimating the value of Orat which the crve near the residual moisture
content becomes vertical. The values chosen are close to the last measured value of the
moisture retention. The resulting van Genuchten fits are acceptable (Figure 42).
In 3 soil samples, select moisture retention data points seemed to deviate from the
general shape of the moisture retention curve (Figure 43). These points are regarded as
measurement uncertainty and are discarded in the curve fitting process.
4.2.4 Las Cruces Data Set
The Las Cruces trench site is a 26.4 m long by 48 rn wide by 60 m deep trench,
located on the New Mexico State University college ranch. It is approximately 40 km
northeast of Las Cruces, New Mexico. The purpose behind constructing this experimental
site was to provide undisturbed soil samples for soil property characterization. Multiple
tests on the physical and hydraulic properties of the soil were performed. Analysis of the
particle size distribution indicate that the soils are mainly sands, sandy loams, loamy sands,
and sandy clay loams (Wierenga et al, 1991).
Of the many samples taken from the excavated trench, unsaturated hydraulic
conductivity measurements are performed on
samples. The conductivity data for these
samples are read off conductivity versus tension graphs provided by Peter Wierenga of the
University of Arizona. Moisture retention and saturated conductivity values are obtained
from the Las Cruces Trench Site Database located on the internet. The address is
61
u(b) 80 cm
Q
,'O
3
.
I
I
I
I
2
00
Q
C
-2
In
1
C
q
>1
Ii
0.
M
0
Q
00
.2
1
0
van Genuchten
Measured Tension
-0
-r0.1
1
moismm content 0
y = vg(. 18, 1.5,01,49)
Value
Error
n
1.5199
0.033581
a
O.OQ876
O.OOW183
Os
ChIsq
R2
0.44631
0.0021862
0.71M
NA
0.94519
NA
Figure 42: van Genuchten Fitted Moisture Retention Curve for WEL Sample ub) 80
CM.
62
IF
u(b) 225 cm
3
I
11 I
I
0
.
.
.
I
I
I
2
3)_
W
.2
C
.2
la
P
1
S
0
M
-
U
00
-Brooks Corey
0
1
0
- Campbell
van Genuchten
Measured Tension
I
-2
I'll
I
0.1
1
moisture content,
y = vg(027,1.5,.01,.48)
a
y = bc(.30044,.1,1,.45122)
IN
Value
Error
y = camp(l,.l,.45122)
11
value
Er-ror
x
0.55632
0.045027
W,
Wh
76.218
8.6571
b
- 11.51
Chisq
0.15002
N
0.94722
NA
Chisq
0.022848
N
R2
0.97449
N
37.44
3.6554
0.81923
Value
0
-
n
a
0
ChIsq
R2
0.30044
1.7724
0.0068872
0.45122
0.0058463
0.99794
Error
0.0688312
0.096068
0.00025951
0.0006644
NA
NA
i
Figure 4-3: INEL Moisture Retention Curve Showing Deviation From General Sh*.
63
:
ftp://meftp.nmsu.edu/pub/soi1s/. Additional moisture retention tests are also performed on
the
soil samples. The values of these additional points are on moisture retention graphs
supplied by Wierenga.
The unsaturated conductivity values are calculated by establishing a steady flow in
the 3-inch cores and measuring the gradient at I cm from the inlet and cm from the outlet
with pressure transducers.
Laboratory saturated hydraulic conductivity values are
determined by a modified version of the outflow method. Moisture retention data is
measured using a pressure cell method.
The moisture retention data.from the soil samples contain the water content at zero
tension. This point is excluded from the optimization process because the use of a
logarithmically transformed scale did not permit a zero tension point. The data point
immediately following the zero tension point is substantially lower -inmoisture content.
Poor resolution of the moisture retention curve near saturation results. The optimization
process for the van Genuchten method is not able to define a reasonable Osvalue. Hence,
Os for each soil sample is fixed at the water content measured at zero tension.
4.2.5 Maddock Data Set
In situ unsaturated hydraulic conductivity experiments were conducted at the Oakes subbranch of the Carrington Irrigation Station during 1972 and 1973. The station is situated
km south of Oakes, North Dakota. The soil found in this region is referred to as Maddock'
sandy loam.
Unsaturated hydraulic conductivity was measured in the field using the
instantaneous profile method. Moisture retention was measured in the lab using a pressure
cell method.
Particle size distribution, bulk density, moisture retention data, and
unsaturated conductivity data can all be obtained from Carvallo et al. 1976). Moisture
retention and unsaturated conductivity values are tabulated for 14 soil samples. The
64
,'Maddock data does not contain measurements of saturated hydraulic conductivity. The
problem encountered with the Las Cruces moisture retention data for the van Genuchten
analysis is also present in the Maddock data. Once again, the value of Os is set equal to the
moisture content at zero tension.
4.2.6 Plainfield Data Set
The
data sets for Plainfield sand are taken from Mualem's soil catalog. The soil catalog
numbers are 4101, 4102, 4103, 4104, and 4105. Moisture retention curves are generated
using only the data obtained from the drainage process. The moisture retention data is
measured using a porous plate setup. Unsaturated conductivity versus tension is measured
by establishing steady flow in a soil column. No model parameters are fxed.
4.2.7 Sevilleta Data Set
The Sevilleta National Wildlife Refuge is located approximately 32 km north of Socorro,'
New Mexico. It occupies an old flood plain area of the Rio Salado, a tributary of the Rio
Grande. Soils in this area typically consist of relatively uniform, unconsolidated, fine
fluvial sand (Stephens et al., 1985).
The moisture retention and hydraulic conductivity data points for 3 soil samples are
obtained from graphs found in Knowlton 1984). The moisture retention values are
measured using the hanging column method. Unsaturated hydraulic conductivity data is
measured using the in situ instantaneous profile method. Saturated hydraulic conductivity
versus depth is measured using a shelby tube permearneter.
Values of saturated
conductivity are chosen by selecting the measured conductivity value closest to each of the
3 sample depths. The moisture retention data is sufficiently detailed, so none of the model
parameters are fixed.
65
Chapter
Analysis of Curve Fitting Results
This chapter discusses the results of the curve fit optimizations. Section one presents a
mathematical analysis of the relationship of the slope of the unsaturated hydraulic
conductivity curves predicted by the Brooks
Corey, van Genuchten, and Campbell
methods. Section 2 scrutinizes the accuracy of the predictive models. Section 3 looks at
the influence of selecting a different match point for the conductivity models. Section 4
focuses on the concept of Leverett scaling.
5. 1 Unsaturated Hvdraulic Conductivity Slope Analysis
The Brooks
Corey and Campbell equations for the moisture retention curve are in the
basic form of a power law. At high tension values, where (aNi)n>> 1, the van Genuchten
moisture retention equation also takes on the form of a power law. By equating the
66
parameters from the moisture retention equation of the three models, an analysis on the
predicted slopes of the corresponding conductivity equations is performed.
5. 1.1 Comparison Between the Brooks
Corey and van Genuchten
Models
At high tension, the van Genuchten moisture retention equation is approximated by the
following power law:
SI
>>
when
(av)-'
By equating the above approximation to Brooks
Corey (Equation 26), the following
relationships result:
a=
1
Vb
X = mn
Assuming that
= mn, the relationship between the conductivity slopes on a log
transformed scale is evaluated by making a simple parameter analysis of the power
exponents. By performing a general binomial expansion,
(I _ Sa)b
I _ bSa
for
S <<
the van Genuchten conductivity relationship with constraint rn =
converts to a power law form similar to the Brooks
(Equation27).
2
K,
-
MSIM
67
- 1/n (Equation 217)
Corey conductivity equation
1
2
M SIM
S'i
for Se << I
M
2
2 m
Se
Equating the power exponents between the two models results in:
2
2
1
3 =-+- 2
Substituting,% = mn into Equation 5. 1, we get
M=-4 I-- 1
5
n
This contradicts our initial assumption that rn = - 1/n and shows that the Brooks & Corey
and the van Genuchten models are entirely different predictive models. In fact, this
analysis concludes that the Brooks
Corey conductivity slope is steeper than the van
Genuchten slope (Table 5. 1).
5.1.2 Comparison Between the Brooks
Corey and Campbell Method
Comparison of Campbell (Equation 21 1) with Brooks
Corey (Equation 26) gives
b
Setting the power exponent of the Campbell conductivity equation (Equation 212) equal to
the power exponent of the Brooks
Corey conductivity equation (Equation 27), the
following equality is observed:
b=A
Brooks & Corey and Campbell have the same slope, but the basic assumptions for each
model differ. First, Campbell assumes that there is no residual moisture content. Second,
there is no specific range of tension for which the Campbell equation is invalid. The
68
Brooks & Corey
M
I
A
van Genuchten
Slopel
Slope2
0.33
0.5
7
6.5
0.5
1.0
5
4.5
0.67
2.0
4
3.5
0.7
3.0
3.67
3.17
0.9
9.0
3.22
2.72
Note: (1) Brooks & Corey Slope =
S,
= 2 +1
M
(2) van Genuchten Slope
=sn =-+= Sk
m 2
2
1
Table 5. 1: Comparison of Calculated Slopes Between the
Brooks Corey and van,Genuchten Models.
69
I
-
2
Brooks
Corey model is valid only when
> Nb and assumes that there is a residual
moisture content.
5.2 Results of the Predictive Models
This section discusses ow the predictive method performs for each soil type. Analysis of
the predictive fit for the unsaturated hydraulic conductivity is limited to data from Cape
Cod, Hanford, INEL, Las Cuces, Plainfield, and Sevilleta. The Maddock data does not
contain saturated hydraulic conductivity values, thus prediction of the unsaturated hydraulic
conductivity curve is not possible.
Brooks
Corey and van Genuchten curve fits are generated for all the data sets.
Predictions using Campbell's method are tested on Cape Cod, UEL, Las Cruces, and
Sevilleta. The Russo model is only fitted to two samples in the Cape Cod data set.
A summary of the overall performance of the models can be found at the end of this
section.
The KaleidaGraphTM generated curve fits for the moisture retention and
conductivity data can be found in the appendix.
5.2.1 Criteria for Fit Acceptability
The criteria for judging the success of the fit depends on the intended use for the predicted
unsaturated conductivity curves. The simple case of water movement through a vertical
profile illustrates this dichotomy in performance acceptance. The slope, dKIde, of a
conductivity curve is typically fairly steep. Data from the conductivity curve can be used in
two ways.
If we assume that the flux through the vertical profile is known, we can use the
unsaturated conductivity curve to predict the moisture content. Since the conductivity curve
70
is steep, a large error in the conductivity prediction does not result in a sizable error in the
moisture content. In fact, the difference between the calculated and the actual moisture
content may be minimal. A relatively large deviation between the measured and predicted
conductivity curve will still be judged acceptable for this application.
The judgment criteria changes drastically if the opposite scenario is chosen.
Assuming that the moisture content is known, we can use the conductivity curve to predict
the unsaturated conductivity value. A small deviation between the measured and predicted
conductivity curve will result in a substantial error in unsaturated conductivity value. Error
in the conductivity can vary by orders of magnitude. This application for the conductivity
curve has a much lower tolerance for error. A predictive fit judged acceptable for the first
case may be judge entirely unacceptable for this case.
This application dependent aspect of the performance criteria dictates the acceptable
amount of error between the predicted and actual values. The performance of the predictive
models will vary from application to application.
In our analysis, a predictive fit was
deemed acceptable if the predictive curve varied from the measured data by less than an
order of magnitude.
5.2.2 Cape Cod
The moisture retention data for the Cape Cod soil is ill defined near the low saturation
region.
During the optimization process, this results in a negative value for
theoretically impossible situation.
To remedy this problem,
, a
, is removed as an
optimization parameter for the curve fit procedure and is introduced as a fixed value.
Current literature (Khaleel, 1995) suggests that a value of 0, in the range of 0 to 003 is
fairly typical for a coarse sand. Hence, the value of 0.01 is selected for the Cape Cod
sands.
71
For the Brooks
Corey, Campbell, and van Genuchten methods, the fitted
moisture.retention curve provides excellent agreement with the measured data. For each
soil core, the curve goes through practically every measured data point. Only one data
point in core 14a noticeably departs from the curve. The Russo model also generates a
good moisture retention fit for cores 12a and 17a.
Although the fitted moisture retention curves are essentially perfect, the predicted
unsaturated conductivity curves deviate from the measured values. All three methods tend
to underestimate the conductivity values for this coarse sand. Predictions using the Brooks
& Corey and Campbell method fall within one order of magnitude for all the samples. In
fact, the Brooks
Corey and Campbell predictions are essentially the same. The van
Genuchten conductivity values vary up to 12 orders of magnitude. Four out of
van
Genuchten predictions fall within one order of magnitude. An three models seem to predict
the unsaturated conductivity reasonably well.
Although the Russo moisture retention fits are acceptable, the predicted unsaturated
conductivity curve are absolutely unacceptable. Values of conductivity are underestimated.
by 20 to 40 orders of magnitude (Figure 5. 1). Given the results of these predictions, the
Russo model is no longer used for further analysis.
5.2.3 Hanford
Fits of the moisture retention curves to the measured data also fare well for the Hanford
soils. Both the Brooks & Corey and van Genuchten models follow the general trend of the
moisture retention data. The sharp turn in the moisture retention data near the saturation
point is relatively well defined by the van Genuchten model. A few of the curve fits
underestimate the rapid downturn, but in general, the van Genuchten curves accurately
represent the region.
72
core 12a
Ocb 6o'O
0.001
1 o-6
0
1 0-9
-12
1Ij
1
w
i 0-15
1 o18
Q
I
o-21
1 o-24
I
2
E
>1
1 o-27
I o-30
o-33
Meaured ConductMty
1
1 o-36
Russo
1 o-39
1 042
1
1
0
Capillary Tension , 4(
Figure5l: RussoPtedictedConductivityCurvefor-CapeCodCore,12a.
73
The quality of the predicted unsaturated conductivity curves fluctuate significantly
for the Hanford sands. The fits for both methods range from acceptable to woefully
inadequate. The unsaturated conductivity data for the Hanford soil is measured by two'
different techniques, the steady state head control method and the ultracentrifuge method.
Measured conductivity values between the two methods deviate drastically within certain
samples. Differences in experimental data can be as high as two orders of magnitude
(Figure 52). Data from the centrifuge samples also register moisture con.tent values that
are higher than the measured porosity (Figure 52). Compared with the steady state head
control values, the ultracentrifuge measurements tends to underestimate the conductivity
values.
Several reasons may contribute to this'variation in measured values. Information
on the physical properties of the soil samples indicate that the measured bulk densities for
several samples differ between the two experiments. Khaleel et al. 1995) evaluate the
effects of density variations on the centrifuge samples and conclude that the deviations
cannot be attributed solely to differences in the density. Another plausible explanation lies
in the possible compaction of the soil samples during the centrifuge process. The'effects
stemming from compaction have not been investigated. Experiments testing this theory
need to be performed. Given the unexplained variations in the centrifuge data and the
proven reliability of the steady state head control method, data from the centrifuge
measurements
is not considered
when comparing the measured and predicted
conductivities.
The success of the predictions also relies on the quality of the measured data. In a
few samples, the measured unsaturated conductivity data are scattered and do not follow a
consistent pattern (Figure 53).
(Figure 54).
In other cases, the measured value of K, is ill defined
Uncertainty in measurement will contribute to the deviation between the.
predicted and measured values of conductivity.
74
^
-
Hanford Soil Sample 0079
^^4
U.U
II
n (MI
I
I
.
I
.
:
I
M
7
CP
0
d
0
1 05
.52
ts
1 o-6
C
0
1 o-7
.S2
75
Ii1
-0
>.
M
0
1 0-
I
0
0
-
8
0
0
C]
SSHC
VanGenuchten
-Brooks Corey
Saturated K ai Porosity -=
Centrifuge
7
n-9
0.1
1
moisture content 0
Figure 52: Hanford Soil Sample 0079. This sample shows (a) the difference
between SSHC and Ultracentrifuge measured data, (b) measured moisture contents
greater than porosity for the ultracentrifuge measurements, and (c) an example of a
good prediction.
75
Hanford Soil Sample 21638
0.01
lz
dU
d
2
z
0
U
.2
M
I
0.001
0
-
0.0001
E]
I
I . . . Iz
SSHC
Van Genuchte
- rooksCarey
Centrifuge
10-5
i 0-6
-2
07
P.,
10-8
09
0
.
I I
0.01
.
I
I
0.1
I
I
.
.
I
.
1
moisture content,
Figure 5.3: An Example of Scattered Conductivity Data for the Hanford Soils.
76
Hanford Soil Sample 21639
I
I
I
. -
.1
10
I
0.001
In
dU
W.
0
1 0-5
13
6
'5
07
I
0
U
.2
:5
co
>1
I
I
10-9
I
-i
i 0-11
I
I
I
.
I I . I
0.01
0.1
I
.
I
.
I
.
.
1
moisture content,
Figure 5A Hanford Sample 21639. This sample shows (a) an example of
conductivity data in which Ks is ill defined,and (b) an example of an unacceptable fit.
77
The predictive curves generated by the Brooks
Corey and van Genuchten
methods are very similar. As shown in the previous mathematical analysis, the slope of the
Brooks
Corey predictions tends to be steeper than the van Genuchten (Figure 5.5).
Good predictions associated with Brooks
Corey model usually implied acceptable
predictions by the van Genuchten model. Figures 52 and 54 illustrate the wide range of
predictions obtained. Figure 52 displays an example of an acceptable prediction. Figure
5.4 shows an entirely unacceptable prediction.
The majority of the Brooks & Corey predictions are within 2 orders of magnitude.
Only one sample falls outside this range. The van Genuchten predictions, minus the same
sample, fall within 25 orders of magnitude. Quite a few of the Hanford samples for both
methods fall within the acceptable order of magnitude.
A general trend visualized from the conductivity curves is a strong tendency in both
methods of supplying better predictions for the fine sand samples. The conductivity curve
for coarse sand samples are consistently underestimated by both methods. As the mean
grain size increases, the deviation between the measured and predicted values also
increases. The Brooks
Corey and van Genuchten methods seem more suitable for
predicting the fine sands samples than the coarse sand samples.
5.2.4 Idaho National Engineering Laboratory (INEL)
As discussed previously in section 42.3, the INEL moisture retention data is altered to
allow a favorable fit to the van Genuchten model. Brooks
Corey, Campbell, and van
Genuchten curve fits are generated for this soil. The fitted moisture retention curves for
van Genuchten correspond well with the measured data. The fits associated with Brooks &'
Corey and Campbell do not agree as well as van Genuchten with the actual measured data.
The unsaturated conductivity graphs contain a measured conductivity point that falls
beyond the predicted curves. We can attribute this outlying point to our curve fitting
78
Hanford Soil Sample 22225
0.01
0.001
Ca
E
U
4
I
0
0.0001
6
1 o-5
0-
06
.2
07
t
0
U
SCa
>1
1 0-8
10-9
I
I
I
I
,
0.01
I
I I
-
0.1
--
, - ---
.
,
,
MMME
, , , -,
1
moistwe content,
Figure
5: The Brooks & Corey Predicted Slope Compared to the van Genuchten
, Predicted Slope.
79
procedure. In selecting the data points used to generate the van Genuchten curve, we
neglect the low tension point near saturation. By doing so, we are allowing the curve
fitting program to determine the optimal value for the saturated moisture content. The
optimized saturation moisture content is usually less than the measured value.
This van Genuchten determined value is then selected as the saturated moisture
content for both the Brooks
Corey and Campbell methods. The measured value of
saturated, hydraulic conductivity is assigned to tis water content. The predictive curves for
unsaturated conductivity are.then generated for the samples. Any measured conductivity
value with a moisture content greater than the van Genuchten determined value appears as a
outlying point. We neglect these points in our analysis.
The predicted unsaturated conductivity curves significantly deviate from the
measured values.
All three models tend to overpredict the unsaturated hydraulic
conductivity. A majority of the predicted van Genuchten conductivities differ from the
measured data by
to 2 orders of magnitude. The Brooks
Corey and Campbell models
display even higher departures from the measured values, falling between
to 25 orders
of magnitude. For the Campbell model, significant deviations between the measured and
predicted conductivity values occur near the low moisture content region. As the moisture
content decreased, deviations from the measured data increased. Predictive fits from all
three methods are unacceptable for the I1,4ELsilts.
5.2.5 Las Cruces
Once again, the measured and fitted moisture retention data agreed well with the Brooks
Corey and van Genuchten models. A less favorable fit is associated with the Campbell
method. The Campbell curve fits agree well with the measured data for samples 425 534, and 72 1, but do not match weHwith samples 334 and 849.
80
For all three models, acceptable conductivity predictions are obtained for 3 of the
soil cores. The remaining two samples deviate between
Brooks
to 1.5 orders of magnitude for
Corey and between 1.5 and 2 orders of magnitude for Campbell and van
Genuchten. Predicted conductivites for the Brooks
Corey method deviate much less
than Campbell and van Genuchten.
5.2.6 Maddock
Moisture retention curves are fitted to the data for Brooks
Genuchten models. The fitted curves for Brooks
Corey, Campbell, and van
Corey and van Genuchten match the
measured data well. The Campbell fit produces a linear line that does not represent the
nonlinear data. Unsaturated conductivity curves are not generated because the saturated
conductivity is not measured. Results from the moisture retention fit are used in the match
point analysis (Section 53).
5.2.7 Plainfield
After analyzing the Plainfield sand, it was discovered that the sand samples are not
representative of the actual Plainfield sand. Each of the
samples have been specifically
sieved to obtain a particular range of grain sizes. These uniform, narrowly distributed soil
samples do not provide moisture retention curves that are indicative of their natural
environment. Thus, the soil samples for Plainfield sand are invalidated from our curve fit
analysis.
5.2.8 Sevilleta
The Brooks
Corey, Campbell, and van Genuchten fitted moisture retention curves
matched the measured data quite well. Excellent agreement is also found between the
predicted and measured values of unsaturated hydraulic conductivity for the Brooks
81
Corey and van Genuchten models. The Campbell model shows excellent agreement
between the measured and predicted conductivity at the high water content regime, but
starts to break down at the low moisture content range. Deviations in the Campbell
predictions are as high as 1.5 orders of magnitude at the lowest moisture content.
5.2.9 Summary
Comparison between the different soil types illustrates a few consistent trends found within
the curve fitting analysis. These trends are defined by textural, rather than structural,
characteristics. A definite correlation exists between the predictive accuracy of the Brooks
& Corey, Campbell, and van Genuchten models and the mean soil grain size. From the
Cape Cod analysis, the Russo/Gardner model appears to be an inadequate model for
predicting conductivity using just moisture retention data. Additional constraints need to
be specified in the Russo model.
There is a direct relationship between the measured and predicted conductivity
deviation and the mean grain size. Using the saturated hydraulic conductivity as an
indicator of the grain size, graphs of the mean error versus Ks are plotted (Figures 56 5.8). The mean eror is defined as:
N
I
mean error =
05i)
N
-
where
8i
=
109 Kpredicted - 109 Kmeasured
N
= the number of measured K data points
The deviation pattern is evident from the graphs. As the,saturated conductivity increases,
the mean eror accordingly increases.
82
Brooks & Corey Model
3
2
&A
1
o
tt=
(D
x
x
0
x
A
x 41
aco
4)
E
A,
AA
A
A
0
X
(EO
0x x x
x
x
-1
x
x Cbcb
OX
xx
0
0.001
0.01
x
CID
-2
x
-3
-4
i
0-5
0.0001
Satumted Hydmuhc Conductivity, 1
o
x
cm/S)
Cape Cod
Hanford
A
INEL
o
Las Cruces
0
0.1
Sevilleta
Figure 56: Mean Error Versus Ks for the Brooks
83
Corey Model.
3
I
Campbell Model
.
I
I
I
I .
I I
2
I
I
A A
A
0)
F TI I
I
I
I
I I
A A
0
0
A
0
0
00
co
E
I
A
C
0
F
A
1
O"
I
,n
AA
AA
I
-1
0
-2
0
-3
-4
1
0.0001
1 . I . 1-
I
I I 1 1 ' . 11
0.001
I
0.01
I
I
.11.
0. 1
ISaturatedHydraulic Conductivity, Ka (cm/s)
o
Cape Cod
A
INEL
o
Las Cruces
0
Figurell:
Sevilleta
MeanEfforVersus& fortheCampbeflModel.
84
van Genuchten Model
3
2
1
P
4)
x
0
XX
aw
a)
E
x
x 4XIx
x
X X
X
OCOD
0
-1
x x
-2
X X
-3
YOX
x
x
11-
-4
i
0-5
11
I
I II
0.0001
lld
I
0.001
0.01
0.1
Satumted Hydmulic Conductivity, 1 (Cm/S)
o
x
Cape Cod
Hanford
A
INEL
o
Las Cruces
0
Sevilleta
Figure 5.8: Mean Error Versus Ks for the van Genuchten Model.
&5
The saturated hydraulic conductivity does not seem to be a reliable indicator for
grain size. The range of saturated conductivity values for the INEL silts is almost just a
wide as the range for the Hanford sands. The inherent differences between the two soil
textures is not adequately represented. A different parameter needs to be selected to portray
the varying grain sizes.
The parameters XVb,We, and
Brooks
are chosen to represent the grain sizes for the
Corey, Campbell, and van Genuchten models respectively.
These three
parameters are a measure of the largest pore size that exists within the soil. The values of
Vb, yfe, and l1a indicate the length of the capillary rise above the water table. As the grain
size increases, the length of the capillary rise decreases. To characterize the trend of
increasing eror with increasing grain size, graphs of mean eror versus the in',---rseof Vlb
and Ve are potted. Figures 5.9 - 5.1 1) show the plots of mez eor vs 11Y(b,mean eror
versus llyle, and mean eror versus a. The systematic departure of the coarser material is
more dramatically characterized by these three parameters than by Ks. A general bias of
underestimating the coarser sands is evident.
The INEL silts significantly deviate from the apparent trend. All three models
consistently overpredict the conductivity values. If the trend is correct, the models should
have predicted the conductivity values reasonably well. An explanation for this devie-,ion
lies in the nature of the soil. The ML
silts are a ighly structured, aggregated soil. Large
blocks of silt aggregate together and act like larger grains. The aggregated nature of the soil
influences the measured saturated conductivity and the measured moisture retention data.
The aggregation may produce soil pores in the INEL silt which are larger than
expected in an unstructured silty material. This leads to higher measured saturated
conductivities. The unusually high range of Ks for this silt reflects the aggregated nature of
the soil. Hence, it is of no surprise that the predictive models overestimate the unsaturated
conductivity.
86
Brooks
3
I I
.
1.
I
I I I I
A
a)
I
I
I
4N &
2
2-
Corey Model
I
A
&-,&
A
I
.
I
I I 1
E
.
.
.
.
I .
i
A
A
A
1
x
0
>;X XX
ffn XX
O'
X
0X0
xxox
w
0
I
Xq30
OX O
-1
X XX
0
XXI
-2
x
.
-3
-4
0.001
I . . .... I
. .
0.01
0.1
1
I /Wb (1/cm)
o . Cape Cod
x Hanford
A
INEL
o
0
Las Cruces
Sevilleta,
Figure 59: Mean Error Versus lWb for the Brooks & Corey Model.
87
Campbell Model
3
A
2
A
A
A
1
2
(D
0
0 0
co
4)
E
-1
00
0
0
-2
-3
-4
0.01
I
I
ill
0.1
1
I
10
We(1/cm)
o
Cape
A
INEL
od
<> Las Cruces
0 Sevilleta
Figure 5. 10: Mean Error Versus 1/yfe for the Campbell Model.
.88
van Genuchten Model
3
2
1
C3
90
0
_
co
0
E
X
x
x
x
AX
XA
x
x
XO ox 000
0
-1
XO
X
0
OX
x
-2
-3
x
-4
0.001
0.01
0.1
1
a (1/cm)
o
x
Cape Cod
Hanford
A
INEL
0
Las Cruces
o
Sevilleta
Figure 5.1 1: Mean Error Versusa for the van benuchten Model.
89
Figures 512 - 514 show the plots of root mean square error versus Illyb, llyfe,
and a respectively. The root mean square error (rms) is calculated by:
(5i2
rms
j=
N
Neglecting the REL data, the Brooks
van Genuchten.
The Brooks
Corey model generally has a lower rms error than
Corey model is slightly better in predicting the
conductivity.
Although the Campbell modellalso has comparable errors, it is not a good predictor
of conductivity at the low moisture content region. The model predicts fairly well at high
moisture contents, but breaks down in the low moisture region. The largest deviations are
located in the low conductivity regime. Since this is the range of interest, the Campbell
model does not meet the acceptance criteria.
5.3 Match Point Selection
The Brooks
Corey, Campbell, and van Genuchten models all ise Ks as the match point
to predict the unsaturated hydraulic conductivity. Recent studies by van Genuchten and
Nielsen 1985) and others recommend that a different match point be used. Use of Ks as a
match point does not make sense for our analysis. Our area of interest on the moisture
retention curve is at the low moisture content region. A match point at K, is a poor
indicator of the behavior of the unsaturated hydraulic conductivity at the low moisture
regime. A more suitable match point appears to be an unsaturated conductivity value near
the region of interest.
Moisture retention values near the saturation point are typically difficult to determine
because of the steep slope. As the curve approaches the saturation point, rapid changes
90
Brooks
3.5
I
I
I
.-
Corey Model
I
11
I
I
I
I
I I
...
3
t0
0
I2
w
Cr
w
-
co
0)
A
A
2
1.5
A
A
AA
A
Cx,
A
A
1.,AAx
# >x
x
x
0 8
A
1
0.5
n
0.001
:
4N
A
E
0
2
X
2.5
x
.
%O
w
x
x
<
xx > xcp(b
O
. . I I . . 11
0.01
x
0.1
I
.
..
1
I 1Wb(Ikm)
o
x
Cape Cod
Hanford
A
INEL
o
Las Cruces
0
Sevill4ta
Figure 5.12: Root Mean Square Error Versus lipb for the Brooks & Corey Model.
91
Campbell Model
3.5
3
0
0
2
tv
2.5
z
2
0
1.5
Cr
9)
C
(a
4N
E
AO
0
A
0
2
AA
0
1
00
0.5
0
0.01
0.1
1
10
I h4f. Okm)
o
Cape Cod
A
INEL
o
0
Las Cruces
Sevilleta
Figure 513: Roe't Mean Square Error Versus II yre for the Campbell Model.
92
.11,
van Genuchten Model
3.5
I
I
I
I I .-
.
.
. I 1 . 1. 1
.
I
I
x
-
3
I.-
2
0
2
w
Cr
w
aw
0
2.5
x
2
.cx xx
A
1.5
A
0
E
0
2
A
1
A
xx
0.5
0
0.001
>OC
zi
xx
I
I
x
. ...
0
x
00
Ax
I A
XxO
x O
OX
x
0
0
x
-
-
I
0.01
0.1
1
a (1/cm)
o
-Cape Cod
x
Hanford
A
INEL
o
Las Cruces
0
Sevilleta,
Figure 514: Root Mean Square Error Versus a for the van Genuchten Model.
93
occur and the curve takes on the form of a vertical line. Small errors in measurement of the'
moisture content in this region can result in large errors in the predicted unsaturated
hydraulic conductivity (Figure 5.15). The value of K, is primarily determined by the
structural properties of the soil. Soil structural properties are characterized as highly
variable in the natural environment.
Given the extreme variability of K, accurate
measurement of its value is difficult to achieve. Uncertainty in the measured value of K is
substantial.
Luckner et al. 1989) derives a form of the van Genuchten equation that allows an
arbitrary match point. Using the Luckner modified method, Yates et al. 1992) and Khaleel
et al. 1995) both show that selecting a different match point near saturation does reduce the
error between the measured and predicted values of unsaturated hydraulic conductivity, but
each reached a different conclusion regarding the success of the procedure. Yates et al.
concluded that the new scaling point does not significantly improve the conductivity
predictions. Khaleel et al. reaches the opposite conclusion.
A crucial assumption implied by this match point theory is that the models
accurately predict the slope of the unsaturated conductivity curve. If the theory is correct,
the predicted conductivity curve is actually parallel to the measured curve. Obtaining the
true curve can be done by simply selecting a different match point. The original match
point, K, , selected by the models is not a sensible choice.
A statistical approach is developed to measure the validity of this theory. Assuming
that the actual slope is predicted by the model, the relative difference,
of the log
predicted conductivity and log measured conductivity for each data point should be exactly
the same. For each soil sample, a graph of either
versus logO or
versus logyf is.
plotted. A linear regression is then performed on the data. If the predicted curve is just an
offset of the true curve, the regression slope, d8 )/d(logO), should be zero. Figures 516
- 5.18 show the calculated regression slopes for each soil sample. Results clearly indicate
94
-150
-125
1:1
2M
3: -100
la
E
.21
I
-75
T.
2
R4A
-50
T
a.
-25
0
0
0.10
0.20
0.30
0.40
Volumetric Water Content
IN
1 -
lo"
1o-2
10,3
Kr
10 -4
i 0-5
i 0-6
10,7
0
-25
-50
-75
-100
-125
-150
Pressure Head (cm of water)
Figure 5.15: Sensitivity of Calculated Hydraulic Conductivity to Saturated Moisture
Content for Uniform Sand (From Stephens, 1985).
95
Brooks
1 0
.
I
1 . 1-
Corey Model
1
.
.
I
I
I
x
5
t+,
A-4t+
0
8-
zC
x > 4
0
XXA
0)
0
XX
0
.(A
U)
P)
+-4*: X
A 0
A
-5
C
0
xX X x
>11:10
0000
5
?
A
2
Ina
-10
-15
0.001
I
.
I
I ....
.
. -
... I
0.01
0.1
.
I I I
1.
1
I 1AVb(1/cm)
o
x
Cape Cod
Hanford
,& INEL
o
Las Cruces
0
Sevilleta
+
Maddock
Figure 5.16: Regression Slope Versus lVb for the Brooks & Corey Model.
96
Campbell Model
10
CD
CL
z0a
5
++
0
<> 4 04-+0
0
W
W
a
cm
-5
2
-10
-15
0.01
1---L-A--L-1
t
0.1
1
I /v.
(Ikm)
0
Cape Cod
A
INEL
o
Las Cruces
+
0
10
Maddock
Sevilleta,
Figure 517: Regression Slope Versus Ilyre for the Campbell Model.
97
van Genuchten Model
I
10
I
''
I
I
I
I
I
I I I '''
.
I I
.II
x
5
I
(1)
CL
z0a
xx X
0
El
4.'O
0+0
1_
x?< x X-
x
0
Ix
A
w
0
A
X
0
W
(3)
C"
XA x
"I 0
C68
A
5
AA
-10
-15
0.001
II
I
I
I
I
I
I I I . I.1
0.01
0.1
I
I
I I I II
1
a (1/cm)
0
x
Cape Cod
Hanford
A
INEL
o
Las Cruces
O
Sevilleta
+
Maddock-
Figure 5.18: Regression Slope Versus a for the van Genuchten Model.
98
that the selection of a different match point is not sufficient. The regression slopes fluctuate
tremendously and do not hover around zero.
A trend is identified from the Hanford soil, the largest data set. Slopes are negative
at low
values and positive at high a values. At the high a end, the predicted
conductivity values are underestimated. A positive slope indicates that the predicted slope
is steeper than the actual slope. At the low a end, the conductivity values that are
overestimated correspond with a negative slope.
This tells us that the predicted
conductivity curve is not steep enough.
Almost all the slopes for the INEL soil reside in the negative region. As mentioned
in the previous section, the INEL conductivities are overpredicted. Once again, the slope
of the predicted conductivity curve is not steep enough for the finer grained materials.
Underpredicted conductivity curves tend to have a positive slope, while overpredicted
conductivity curves have negative slopes. This implies that the largest deviation between
the measured and predicted conductivity values occur at the low moisture content region.
The value of the regression slope relates to the error in the exponent of the,
predictive conductivity equations.
Assuming that both the measured and predicted
conductivity curves are defined by simple power laws, we get:
K
K
measured
C.
(5.2)
ON
(5.3)
cpgN+S
predicted
where
Cm, Cp
= constants
N
=
the actual conductivity slope
N+S
=
the predicted conductivity slope
Dividing Equation 52 into 53 and performing a logarithmic transformationresults in:.
99
Kpredicted
C
K measured
C.
Os
(5.4)
log Kpredicted= Slog(O) + log
C.
Kmeasured
(5.5)
The regression slope is defined as:
regression slope=
d3
d(log 0)
where
3
log K predicted - log K measured.
Taking the derivative of equation 5.5 gives
Kpredicted
d log K measured
=S
d(log e)
d(8)
d(log 0)'
Hence, the regression slope is equal to the eror in the exponents. It is interesting to note
that the eror between a single measured and predicted conductivity value is influenced by
two different factors: the error between the conductivity curve slopes, S, and the value of
the moisture content, 0, of interest (Equation 54). As you move further away from the
match point, the eror steadily increases. The deviation between a single measured and
predicted value has absolutely no dependence on the actual slope, N.
Regression slope values range between 15 to 10. These are significant errors.
Selecting a match point at coordinates (01 , K1 ), Equation 54 becomes
K predicted
e S
Kmeasured
19,
100
Assuming a match point moisture content of 03 and a regression slope of 3 the error
between the predicted and measured conductivity at
K
predicted=
K measured
0.05
3
=
0.3
1
=
is
3
6
The predicted conductivity will underestimate the measured value by a factor of 216, well
over two order of magnitudes. This shows why the saturated conductivity value is not
considered to be a good match point.
5.4 Leverett Scaling
Pore space within soils may be visualized as a series of capillary tubes or circular rods.
Given this idealized assumption, a single relationship between the capillary pressure and
saturation can be derived for similar soil types. This simplified concept suggests that the
moisture retention curves of different soils can literally be reduced to a common curve by
selecting an appropriate scaling factor. Based on fluid and medium properties, the scaling
factor normalizes the relationship between the capillary pressure and saturation for each soil
type.
Leverett 1941) employs dimensional analysis to derive a semi-empirical equation
which establishes this constant relationship between tension and saturation. This equation
is known as the J-Leverett function:
= J(S.,)
(5.6)
where
P
=
the capillary pressure
a
=
the surface tension
101
k
=
the permeability of the medium
=
the porosity
A consistent system of units must be used in defining the J function.
The theory behind the J-Leverett function is that there is some dimensionless
relationship that exists between capillary pressure and saturation. The van Genuchten
moisture retention equation specifically relates the capillary pressure to the saturation. If
we non-dimensionalize the van Genuchten equation by using a scaling factor, the resulting
expression can be regarded as a, J-Leverett function. The key parameters in van Genuchten
that influence the general shape of the moisture retention curve are a and n . Figure 519
shows the relationship between a and n for the various dat. sets. A slight trend is
observed with the Hanford soils, as a increases n decreases. The value of n is affected
by the soil grain size.
Figure 520 (a) displays the influence of n on the moisture retention curve
expressed in the dimensionless form aV versus Se and clearly shows that n has a strong
influence on the curves in this form. In keeping with the Leverett concept, an approximate
single non-dimensional form of the van Genuchten relationship can be obtained by scaling
the moisture retention curve. The van Genuchten equivalent of the J-Leverett function is
obtained by scaling the value of aV by its value at some selected effective saturation value,
11N:
J(S' = a
(av
-I
(5.7)
=
N
where
102
5
I
I
1111
I I
I
I
4.5
0
4
0
I
I
I I --
0
3.5
3
2.5
x
0
xXA
2
X< A A
1.5
A
0a
x
x
A
x
0
. &Y6A
&L
I
1
0.001
I I I
I 1
I
I
I I I . I. 1
0.01
0.1
% Z
X*
xx
A
I
I
I
1
a (1km)
o
x
Cape Cod
Hanford
A
INEL
o
Las Cruces
0
Sevilleta
Figure 519: Graph of n Versus a for the van Genuchten Model.
I
103
, -4
I U,
1000
100'
aw
10
1
n
0.1
S
n
n
n
n
= 125
= .5
= 175
=2
n =2.5
n=4
Figure 5.20 (a): Graph Showing the Muence of the van Genuchten parameter n on
the Moisture Retention Curve.
104
N
=
the inverse of the matching effective saturation value
(ayf) i = the value of ayl at an effective saturation of IIN
N
Figure 520 (b) shows the scaled moisture retention curves at a matching effective
saturation point of 13. The scaled moisture retention curves do not scale onto a single
curve. Moisture retention curves with varying values of n do not have the same shape.
The slopes between the moisture retention curves are different. The use of a scaling factor
does not alter the slope of the original curve. The scaling factor just scales the curves to
intersect at a common point, the selected effective saturation value. If the moisture
retention curves are to collapse onto a common curve, modification to the original slope of
the moisture retention curve must occur. Therefore, we can conclude that Leverett scaling
is not valid over a wide range of n values.
Although the scaled retention curves still do not fall onto the same cune, Leverett
scaling appears to have some merit over a narrow range of n . With the exception of
Sevilleta, each individual soil used in our analysis falls within a fairly narrow range of n .
The value of n for the majority of the Hanford soil is between 14 to 19. Hence an
analysis of Leverett-like scaling on these soils may be performed.
By substituting the relationship
Ks
I-a
-6
where
Ks
=
the saturated conductivity
9
= gravity constant
P
=
the fluid density
105
-I
I
UU
100
1 0
aw
(ay),/3
1
-C-d
0.1
0.01
n MI
0.1
Se
n
n
n
n
= 125
= .5
= 175
=2
n =2.5
n=4
Figure 520 (b): Scaled Moisture Retention Curves for Various Values of n at an
Effective Saturation Match Point of 13
106
=
V
kinematic viscosity
and Equation 57 into 56, the following equality is observed:
C
where
C
a constant
We have assumed that the surface tension is constant. If Leverett-like scaling holds over a
narrow range of n
graphing a versus K for each soil type on a log transformed scale
0
should show a linear slope of 0.5. As seen in Figure 521, the calculated slopes for all the
soil types, except Sevilleta, are quite close to 0.5. Although most of the slopes exhibit a
Leverett-like trend, the data points do not all fall onto the fitted straight line. The data
points are quite scattered around the linear line. For Cape Cod, R4EL, and Las Cruces the
deviation from the line is fairly small. The Hanford data displays the largest amount of
scatter. Though the data points exhibit a large amount of uncertainty, a general trend is
observed from the Hanford soil. We see a corresponding increase in a as K increases.
0
Evidently Leverett scaling does represent the general trend i capillary pressure
characteristics of the soils investigated here, but certainly does not capture much of the
variability seen among individual samples. The indication is that Leverett scaling will not
capture the natural variability of capillarity encountered in field soils.
107
---I
1
I I-
I I Tf-
-
)'
' ' '' I
-
I
x x
x
x
0.1
l---l"Cm
A
x
7
I I
0.001
0.0001
,
I I I -I
-.1,
A
A
A
I.X
- &- - INEL
- - 0 - - Las Cruces
- ] - - Sevilleta
<
IL
0
El
A
I
A
I
I
I I - II
I
I I I . ''.I
0.01
0.001
Cape Cod
- Hanftrd
1-1
x
-I -V
K/
e
X
(9-x-
0-0
0-- - -;--910 >' x x
0.01
I
>0<
x
-
I I
I I I I I
0.1
1
(cnVs)
y = 085562
-
I
xA(0.6348) R2= 047459
- y= 1.5362'xA(0.60251)R2 0064162
-
- y = 043918
XA(0.58286)R2= 035023
- - - - - y =057"7 xA(0.3596)
R2 063157
- . - .
y = 65565 XA(l.7343)
R= 09995
Figure 5.2 1: Graph of a Versus
K
zzL..
0
108
Chapter 6
Conclusions
6. 1 Summary
The purpose of this thesis is to critically assess the reliability of unsaturated hydraulic
conductivity models that predict conductivity using only moisture retention data and
saturated hydraulic conductivity. Specifically, the focus is on the predictive ability of the
Brooks
Corey, Campbell, and van Genuchten models in estimating conductivity values
for aquifer like materials. Soil samples from 6 distinct sites are analyzed. The soil textures
range from coarse sands to fine silts.
Soil in the unsavuratedzone of the aquifer resides in the low moisture regime of the
moisture retention curve. Accurate prediction of the unsaturated conductivity is crucial
when modeling contaminant transport through the vadose zone or recharge to the aquifer.'
The relevant range of conductivity values associated with the unsaturated zone is typically
orders of magnitude smaller than the saturated conductivity.
Thus, the analysis
concentrates on the low moisture content region that is characterized by low unsaturated
conductivity values.
109
Results from the analysis indicate that the predictive methods generally do not work
in predicting the unsaturated hydraulic conductivity. There is no supporting evidence that
shows that these models accurately characterize the natural variability in the aquifer.
Although the analysis concludes that the models are inadequate, there are a few interesting
trends represented in the data.
•
A direct correlation exists between the predictive error and the mean grain size.
Deviations between the measured and predicted conductivity increase as the
texture of the material becomes coarser. The models predict conductivities for
the fine to medium sands fairly well, but consistently underpredict the coarse
sands.
•
The Brooks
Corey model is slightly better at predicting sands than the van
Genuchten model.
•
The Campbell model predicts well in the high moisture content range, but
breaks down at low moisture contents.
Recent questions have been raised by researchers regarding the selection of the
saturated conductivity as the match point for these models. If the area of interest on the
conductivity curve is in the low moisture content region, then using the saturated
conductivity as a match point for the predictions may not be adequate. Investigators
suggest selecting an unsaturated conductivity value near the region of interest as a plausible
match point. An inherent assumption in this theory is that the slope of the predicted
conductivity curve reflects the actual slope. A statistical analysis is performed to evaluate
this theory. The conclusions are:
•
The slope of the predicted conductivity curve does not represent the actual
slope.
•
Predicted slopes of the coarse sands are steeper than the actual slopes
110
•
Predicted slopes of the fine sands are not steep enough.
•
The total error between a single measured and predicted conductivity value is
determined by the difference between the predicted and measured slope and the
value of the moisture content of interest. As you move further away from the
match point, the error between the values increase.
In numerical modeling of heterogeneous soils, the capillary pressure curves are
often scaled by the spatially variable saturated hydraulic conductivity such that a single
curve represents any point within the aquifer. The concept of Leverett scaling has been
used to represent spatially variable moisture retention and relative permeability
characteristics (Keuper, 1991). Conclusions on our analysis of J-Leverett function are:
•
The soils from each site do show a general trend of the van Genuchten
parameter a increasing as the square root of the ratio of saturated conductivity
and porosity, as implied by Leverett scaling.
•
The natural variability in capillarity among individual soil samples from a site is
not captured by Leverett scaling.
6.2 Future Research
Several aspects of the work in this study can be continued. First, hysteresis of the
moisture retention curve is not addressed in this analysis. The ephasis
of the study is on
the drainage cycle. Research on the predictive ability of the models using the wetting cycle
values may prove interesting. Second, the Brooks
Corey and van Genuchten models
both contain predictive equations for the nonwetting phase. Data sets (Demond, 1988 and
ill
TerraTek, 1994) of the measured unsaturated conductivity of the nonwetting phase do
exi,,7, -inally, the concept of Leverett-like scaling does warrant some additional research.
The soils at each site do reflect the general trends in capillarity implied by Leverett scaling,
but Leverett scaling does not represent the variability seen among individual samples at a
site. Perhaps a modified form of the J-Leverett function is the solution.
112
References
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1
Fetter, C. W., Contaminant Hydrogeology, 458 pp., Macmillan Publishing, New York,
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116
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118
Appendix A: Table of Moisture Retention
Parameters
119
I
0
0
m
m
M M
CO) kon
r,M
co0W
CM r-
Iw
r,
w
N
CYN00w
oN
N V
r--cm
to co
m v
0
N
m
v
w
8
COM
I
N
N
N
w vN
N
v
W
0
0
N
m
0
M r-
0
N
r-0
-
th w
N w
N
w
U Ci C49
9 "I
C-4
f,0
f
I-v
N
r
w
08
rCvCT
P.
CM
0co
GO
CM co
ILnn
V
M W
0
V
C.,w
P- in
P.
W q6
M
N
co
m
w
W M N
in
d 0 0
N
I-
Iw
A G
co
N - - Ip
- -q -p -Rq- - - - - q- q
- PI:
- -. - . - .
0
V)
Cm
C-* 0
CM
CO
m
le
q 6 6 6
N
Ln
0
0 0 ci
(O
Ln
M
U')
in
C"
N
w
CM
U') VI
q
1
N
N
CD
a
a
10
"It
0 %n
co
0
" cm ;; C,-"I
Ci
C4
C4 C4 C\iCqN
to cm
ci
Cs
to
N - - - "It
-
C
cmNco
r,0m
Ln C* PIN
m
w tom
inN
9
Ln
N
N
Nm
N m
0 N N VM o 0 M Q
0
cnN
m tn
m
CD
0 V f-
cm
N
co tv
C4 .
a
r- co
C 6 C66ci d6
d C;
Ln
Ln
0
in t.
0 0 0
.
co
Csci 6
Cs
C
>
0
C
cm CD0
f- r- co
0
0
0cmm N
cm
N
m
0
0 m0 0
-Z
0
'E
LU
C
cm cm
I
120
I
T
co N
3 -
co COV
r- CD
WOWR-RN
wvvvv
M
t- 0
M P- w
0- m w
co
V
to M M
0
- M
co vIV 0 cm
- co
0 vco
N
co
LO
W w
MIO-CMON-92
vMMMMmv
"-4 NM (0
w
M
N
co
M N 0
W co
LO
M V w
N cm0
V rl
to
co
r-
co
In
C
to
.2 li Ci
v- - -
M v P r_
co
co m r.
.. N
0
0 co
w
N
IT
Ln
M
-M -M
M
- - . -IT rl
- -q -q - - -N - -
vM W
M
vLO
"R '10:
. p p C4 In r. C!
V - - MN N -
M
N
M
-co
m
P": "R -
cm
- - N M
M 4
C
C
N
aM v N
N M
CY co
w N
N
tn M W -
M W N
C
N
coo
C
10
0c
-E Q LnOwwwwwIn
v v v- vPA
Pi
9
-01
co z
'5 w w 51
m .0
:cr w
Cf) Cl Cl) N V
4
Lb V-
b
CY
I
r
.0 co
C? C? IT W) co
cm Cy cm
(D
U
M M m M w
7
IV
in in
121
w w
w
I 10 I" 1'* I'm
1
CD
I-I
Data set
sample no
Brooks Corey theta r
Brooks Corey lambda
Brooks Corey psi b
0 Hanford
1
2
3
4
5
6
7
a
9
10
11
12
13
14
15
16
0-072
0-079
0-080
0-083
0-099
0-107
0-113
1417
1419
1636
1637
1638
1639
2225
2226
2227
2228
0.042312
0.054027
0.027662
0.022778
0.018926
0.013680
0.016597
0.030052
0.0066996
0.014483
0.010847
0.0051153
0.0099868
0.018334
0.012324
0.014706
0.0095244
0.78400
1.0445
0.91498
0.46927
0.45506
0.54280
0.66471
0.65821
0.38W1
0.47367
0.45719
0.39605
0.41674
0.76590
0.39818
0.71812
0.49983
79.597 94.761
86.948
102.79
41.345
4.7178
13.355
132.74
1.7186
3.2179
3.8878
14.312
1.3502
15.187
1.1562
8.0089
1.8993
17
18
19
20
21
22 Cape Cod
23
24
2229
2230
2232
2233
2234
12a
13a
14a
0.012452
0.034990
0.012761
0.0071451
0.0080274
0.0100000
0.0100000
0.0100000
0.50517
0.52467
0.49613
0.37914
0.48955
0.86425
1.0024
0.94350
2.2607
59.562
7.3941
1.9M
11.954
2.8342
3.0247
2.9443
25
26
27
28
29
15a
16a
17a
u(a) 30
u(b) 30
0.0100000
0.0100000
0.0100000
0.21804
0.21241
0.92081
0.99606
0.93274
0.25230
0.57177
3.6077
4.1004
2.9482
39.053
27.758
d(a) 30
d(b) 30
U(a) 80
u(b) 80
d(a) 80
0.20605
0.19438
0.18255
0.18275
0.13970
o.85273
0.58358
0.303372
0.56172
0.33613
70.821
55.944
26.484
20.011S I
22.524
30
31
32
33
34
NlmmoAdaho
122
Data set
sample no
Brooks Corey theta r
Brooks Corey lambda
Brooks Corey psi b
35
36
37
38
39
40
41
42
43
44 Las Cruces
45
46
47
48
d(b) 80
u(a) 145
u(b) 145
d(a) 145
d(b) 145
u(a) 225
u(b) 225
d(a) 225
d(b) 225
3-34
4-2S
57-21
8-49
0.18362
0.26633
0.18930
0.13724
0.23377
0.22335
0.30040
0.18619
0.18229
0.054744
0.040695
0.0000
0.044061
0.059871
0.40873
0.46938
0.27584
0.25311
0.50226
0.39026
0.55632
0.49396
0.45317
0.46684
0.28030
0.23908
0.19510
0.36380
26.081
.29.354
34.801
5.1503
16.140
38.517
76.218
20.280
13.595
3.6048
7.8747
6.0231
9.5726_
4.4293
49
50
51
52
30.5
61.0
91.5
1-1
0.042970
0.050779
0.061856
0.10464
1.6349
1.6976
1.8389
0.20278
23.522
21.086
22.467
12.175
53
54
55
56
571
58
59
60
i-2
1-3
1-4
1-5
1-6
1-7
2-1
2-2
0.083099
0.11662
0.11771
0.069635
0.12449
0.10760
0.16921
0.11447
0.33460
0.43695
1.0399
2.0509
1.4714
1.7706
0.37822
0.40180
12.025
12.639
14.980
19.765
15.720
12.336
13.767
14.695
61
62
63
2-3
2-4
2-5
0.10937
0.096538
0.093227
0.42027
0.93136
1.8406
16.144
14.917
18.804
64
65
2-6
2-7
0.088395
0.12680
1.9485
1.4980
16.586
11.351
Knowlton
Maddock
123
Data set
Campbell psi e
Campbell b
12a
2.7801
1.2391
1
15a
3.5445
1.1602
2
17a
2.8569
1.1611
0
3
Cape Cod
INEL
sample no
u(a) 30
20.234
13.785
4
5
6
7
u(b) 30
d(a) 30
d(b) 30
u(a) 80
5.2978
28.038
23.952
17.828
8.7186
5.8580
6.2032
6.9436
8
u(b) 80
1.8582
8.4361
9
d(a) 80
13.668
6.0857
10
11
d(b) 80
u(a) 145
13.240
7.0915
7.1098
12.661
12
U(b) 145
19.144
10.519
13
d(a) 145
3.4767
6.6515
14
d(b) 145
10.137
5.8214
15
U(a)225
20.912
8.4134
16
u(b) 225
37.440
11.510
17
d(a) 225
14.013
5.7495
18
d(b) 225
19 Las Cruces
20
21
22
23
24 Sevilleta,
25
26
27 Maddock
28
29
30
3-34
4-25
5.34
7-21
8-49
30.5
61.0
91.5
1-1
1-2
1-3
1-4
31
8.1387
6.1479
0.95922
5.6952
6.0231
8.1023
1.8623
21.916
18.669
19.826
10.695
9.8333
8.1663
5.5165
4.7307
5.1302
4.1826
6.8135
5.3017
1.0442
1.1321
1.1759
7.5128
4.5437
4.6785
3.5225
1-5
9.6942
1.8798
32
33
34
35
36
37
1-6
1.7
2-1
2-2
2-3
2-4
8.2134
5.8964
9.4994
11.176
12.470
6.7066
2.8173
2.4743
6.6670
4.7752
4.6204
3.1787
38
2-5
10.570
2.0341
39
40
2-6
2-7
8.5554
3.2078
2.1139
3.276
124
Appendix B: Cape Cod Curve Fits
125
core 12a
1
0.95
to
.2
0.9
CF
w.2
C
q
0.85
i.11
'a.
0M
0.8
to
.2
0.75
0.7
0.1
1
moisture content,
I
=
Value
x
0.86425
wo
2,8342
Chisq 0.00036602f
-
R
y = Carnp(l,.11,33)
- Value
Error
0.032252
0.097262
NA
0.996541,
IV,
2.7801
0.095759
b
1.2391
0.045459
1 Chlsq
0.00035419
I
R1
0.996651
NA
F
0.1
y = vg(0.01,1.5,.CO1,.33)
--- VaFue
Error
Error
n
a
T --
- R 1
I
-
I
I
0.9971
I
I
I
I
t
In
4
0.01
0
.a:
U
I=
0
U
.2
75
i!
>1
0
0
0
0
0.001
K (cnVs)
0
-
-
-
- BrooksCorey
Campbe.1
Van Genuchten
I
0.0001
0.1
I
1
moisture content,
126
0.047805
0.0080068
Chisq 0.00030591-
NA
NA
-----I
2.1163
0.25223
NA
NA
core 12a
-0.7
-0.75
04
0
-0.8
2
0Ij
0
-0.85
2
-2
0
E
to
.2
-0.9
-0.95
10
1
Capillary Tension,
y = rus(.01,.l,.l,.33)
b
C
Chisq
Value
12.436
102
0.0003783
Error
12.465
111.36
NA
R
0.994471
NA
0.001
-F--
-'--
Oobdo'O
0I
1 o-6
09
In
U
1 o12
2
i 0115
1 o18
o-21
U
I
C
1 o-24
U
1 o-27
0
.2
E!
>1
1 030
1
0
o-33
1 o-36
-
-
Meaured Co nductivity
- Russo
1 o39
1
I
I
-42
I
,\
,
,
I
10
1
Capillary Tension
127
core 13a
10
-
O&
Q
ci
.LI
CA
54
>1
g
-
- BrooksCorey
E)
van Genuchten
r-
-G-
1
0.
0U
to
.2
I
0.1
I
I
.
0.1
1
moisture content,
y = bc(.01,.1,1,.337)
Error
0.053053
x
1110
1
hisq
I
R1
y = vg(0.01,1.5,.001,.337)
Value
1.0024
3.0247
0.14388
0.000728921
NA
0.993081
NA
Value
n
2.2586
(X
0.24737
Chisq 0.0007469
R
Error
0.082065
0.011871
NA
0.9929
NA
I --1If:
0.1
10
U
4
0.01
o
0
0
U
0
0
U
.2
0
0
0
0
0.001
0
.It
73
-
x
K (cm/s)
Brooks Corey
Van Genuchten
0.0001
0.1
1
- moisture
128
content,
core 14a
1
0.95
30to
0
ci
0.9
0
C
>1
1.
.0.85
.3
'S.
M
U
0.8
Mc
0
0.75
0.7
0.1
1
moisture content,
y = bc(.0l..1,1..353)
x
IVA
Chisq
R
y = vg(0.01,1.5,.001,.353)
Value
0.9435
Error
0.060046
2.9443
0.16436
0.00131851
0.990041
n
a
hisq
NA
R 1
NA
0.1
w
I
Value
2.22
0.24554
0.000928531
ErroF
0.080748
0.01135
NA
0.992991
I
I
Njj
I
.
--
0.01
U
0
Id
0
0.001
U
M
0
U
0.0001
U
75
M
1.
0
-
05
I
K (crTVs)
Brooks Corey
Van Genuchten
1 06
0.1
1
moisture content 0
129
-
core 1Sa
1.15
I
1.1
to
.2
ci
0.2
r.
q
>1
q
0.
14
1.05
I
r---Ir--r-
-
- BrooksCorey
- Campbell
E)
van Genuchten
-
1
0.95
0.9
U
O
0
0.85
0.8
0
nv. 17r
Ij
0.1
1
moisture
y = vg(O.01.1.5,.001,.351)
= camp(1,1,351)
Value
y = bc(.01,.1,1,.351)
-- 75-ue
ontents 0
Error
E For
x
0.92081
0.050034
W,
3.5445
0.16498
wo
3.6077
0.0011654
0.17011
b
1.1602
0.061005
Chisq
R
NA
NA
0.99271
0.1
0
U
Chisq
R
0.0010934
0.99316
Value
n
a
Chisq
NA
NA
R
2.2107
0.19658 i
0.000599i2 1
0.99626 I
11
I
I
I
I - -- 1
r -- :
0.01
-7
d
A
.tI.
U
lz
r
0.001
0
.2
o
U
Fi
-
-
0.0001
-
- Campbell
:
Van Genuchtan
I
I
05
K (crTVs)
- Brooks Corey
0.1
I
I
.
.
1
moisture
130
ontents 0
Error
0.058968
0.006499
NA
NA
core 16a
1.1
I
I
T----- -
I
I
I
I
1.05
--
- BrooksCorey
E)
van Genuchten
--C4
0
1
ci
.2
CA
-
2
0.95
0.9
M
U
b4
q
0.85
0.8
0.75
I -- -- I
I
I
0.1
1
moisture content,
y = bc(.01,.1,1,.348)
1
I
y = Vg(0.01,1.5,.001,.348)
x
Value
0.99606
Error
0.043396
wo
4.1004
0.13704
Isq
0.000742461
R 1
0.995291
NA
NA
Value
n
2.3956
(x
0.16834
Chisq 0.0002596
R
0.99836
Error
0.045i62
0.0029993
NA
NA
0.1
14
2U
4
0.01
.:t
5
V
0
U
.2
75
M
w
0.001
0.0001
0.1
1
moisture content,
131
i
core 17a
1.1
.
.
T_
I
I
F
.
I
I
I
I
i.05
1
20
0
CF
0.95
.2
_
14
0
0.9
.a
0.85
M
U
to
-
0.8
.2
Brooks Corey
-
-Campbell
E)
, van Genuchten
0.75
0
0.7
I
I
I
I
0.01
I
I
0
I
0.1
1
moisture content,
y = bc(.01,.1.1,.356)
Value
C
Value
0.93274 0.12449
d
2.9482
, Chisq
I
y = camp(l, 1,.356)
Error
0.41483
-0.0084693,
R 1
a
0.958271
1
U
1.1611
0.15168
Error
2.1054
C
0.26253
0.034451
Chisq
0.0068349
NA
NA
0.959881
NA
R
0.966471
NA
-
6
-
I
I I I
a
K (CrrdS)
I
Brooks Corey
- Campbell
Van Genuchten
0
0
0.001
_
0
U
.2
17
0.0001
1
0-5
0.01
0.15769
NA
-
.d
b
b
0.0081494
0
0.01
Value
2.8569 0.40507
NA
0.1
0
y = vg(0.01,1.5,.001,.356)
Error
I
I
I
I I I I I I -
0.1
moisture content,
132
I
I
-1
I-
I
.-
1
core 17a
I
-0.7
I
. . I
6
I
I
I
.
.
I . r
-0.75
0to
0
a
-0.8
-
-0.85
S
0
0
7
-0.9
a
R
C
00
-0.95
0
-1
-1.05
100
10
1
Capillary Tension,
y = rus(.01,.l,.l,.356)
a
m
Chisq
Value
114.83
1013
0.0030654,
Error
1208.2
10793
NA
R
0.978131
NA
0.012
-0
0.01
fj
4
;;
fj
C
-
0
0.008
0.004
73
m
S.
>1
0.002
Measured Conductlvfty
Russo
0
0.006
0
U
.2
0
0
0
0
-0.002
5
6
7
8
9
Capillary Tension
133
10
11
2
3
Appendix C: Hanford Curve Fits
134
0-072
5
I I . 11
I
I
I
I
I
I
I
I I I I I
I
I . . I .
4
D-
3
W
0
d
.2
In
a
S
2
1
'a.
M
0
0
to
.2
9
-
.1
van Genuchten
- Brooks Corey
-
0
I
-2
0.01
I
0.1
1
moist= content,
Y= vg(0.01,1.5,.001,.3275)
a
0
Chisq
F32
Value
Error
0.045513
0.0036503
2.0967
0.0061631
0.3068
0.0024439
0.32001
= bc(.01..I,1,.5275)
-VaTu- Error
et
0.042312
0.0023744
X
0.784
0.07719
Wb
79.597
12.269
ChIsq
0.36548
NA
R2
0.95975
NA 11
0.00014252
3.6257
NA
0.85409i
NA
0.001
I
I
I
.
I
I
I
i 0.0001
0
d
i O-Si
LI
/I.
.2
03
C
0
i 0-6
0
07
C.)
r
0
r
0 0
0
13
0
0
0
E
x
CI
13
/ OF
I
1
n-8 r
I
SSHC
Van Genuchten
0
-
W
1
0
O
- BroolmCorey
Cenftigo
-i
09
0.1
1
moismmcontent a
135
0-079
3
D_
04
2
.2
ci
.2In
C
1
2
S
0
M
U
00
.2
1
-2
0.01
0.1
1
moisture contem
y = vg(O.01.1.5,.001,.3754)
Value
Error
0
a
Ob
Chisq
R2
0.047929
2.4132
0.0058319
Value
0.28107
3.1945
0.0057308
0.36001
0
0.00069162
3.7594
0.68376
y M bc(.01,.II,.3754)
0.001
U)
i
le
X
1.0445
0.18384
,Yb
94.761
8.08
R2
NA
I
0.014077
Chlsq
NA
Error
0.054027
I
'
0.0071139
NA
099045
NA
I
.
I :
I.0,
0.0001
a
IF
0
i 0-5
.n
.2
Ts
1'0-6
C
0
C.)
1 0-7
.2
r
-
x
Lj
r
0
0
0
e
'O
0
08 r
0
I
13
SSHC
Van Genuchten
-Brooks Corey
Cenffl"
7
09
0.1
1
moistwe conten
136
0
0-080
I
5
I
I
I
I
,
,
.
.
.
I
I
,
I
I
.
.
I
I
V--T
.
.
I
.
.
4
za
a
3
li
C,
.'A
2
q
1
'EL
Q
co
Q
e-
0
-1
I
-2
I
0.01
. .
0.1
1
moisture content,
y = vg(O.01 1.5,001,375)
Value
0.025562
1.964
0.0074827
0.37
0
n
Cx
0
Chisq
NA
I
0.01
Q
:4,
I
'yb
.1
.NA
-11
I
Value
Error
0.027662 0.0015366
0.91498 I 0.099997
86.948
18.367
8
Chisq
i
0.1
I
3.561e05
0.73307
0.97081
R2
I = Oc(.01,1,1,3669)
Frror0.0035078
0.12308
0.0013148
0.25812
0.95933
I
0
I
I
I
I
NA
NA
.
I
z
F
.-I
r
0.0001 r
0.001
-i
-i
0
z
0
U
.2
E!
>1
05 r
0
-i
06 r
07 r
0
0
-1
- BrooksCorey
0
I ____
09
-i
I -
Van Genuchten
0
r [3
08
SSHC
I
0.1
Centrifuge
.
-i
- --L ---- L--A-
1
moiswm content,
137
0-083
5
I
1
I
I
II
I
I
I
I
I
I
4
BO-
.2
3
1i
.2
2
20
1
'E.
In
Q
0
0
I
v
-
-
Vail %.;lfjllUk;FlUJFl
- Brooks
Corey
3
-1
-2
0.01
I
I
I
I I
I
I
0
I
I I
I I . . . .
0.1
1
moisture content,
= g(0.01,1.5,.001,.3487)
of
Error
0.021111
n
a
0
1.8046
0.0042779
0.34002
0.34248
0.0016949
0.00017802
Chisq
3.2456
NA
Chisq
0.855491
NA
R2
0.001
U)
i 0-
Zs
1 o-6
00
.2
p
'a
I
I
Err Or
0,
0.022778
0.0066166
x
W,
0.46927
102.79
0.036067
8.
0.023153
N
0.99465
I
I
I
NA
I
I
I
:
0
0.0001
1-11
'aC
y = bc(.01..I.l..3487)
Value
Value
0.0473
0
0
0
1 o-7
0
08
0
0
-
0
C3
SSHC
Van Genuchten
-Brooks Corey
Centrifuge
09
0.1
I
moisture contem
138
0-099
5
1
I
.
I
I
.
I -
I
I
I
I
I
_T_ I
Ia
4
co
q
ci
3
.2
In
2
_
14
1
'S.
M
U
0
0
G
ve
-
BI
-
-1
-2
I
I
I
-
I
.
.
I
I
I
.
-
-
I
.
_LLj
0.1
0.01
I
moisture content,
I
I
y = vg(O.01.1.5..001,.3375)
MI
Error
0.03271
Or
n
Y = W-01,10,3375)
Value
0.0055012
1.6567
Or
;L_
WA
0.085824
0.011548 0.0018096
0.33 ' 3.31490-05
0.33276
NA
as
Chisq
R2
0,986891
Chisq
R2
NA
11
0.018926
0.45506
41.345
Error-
0.011292
0.073243'
7.5924
0.32981
NA
0.97256
NA
I
0.001
__T__
.
.
.
I
.
I
I
-
I
.
.
.
.
.
-
I
w
0.0001
d
d
a
i 0-5
SSHC
Van Genucmw
-Brooks Corey
CentrIfuge
0
I
- -0
i 0-6
00
U
.2
1 o-7
a
i o-8
0
I
09
0.01
I
I
.
. . . . .I
0.1
moisture content, 0
139
. . . .
I
1
0-107
5
1
.
I I I
4
3-
3
.2
Ci
.0
C
q
2
.2
1
t,
U1%
0
00
.2
-1
-2
0.01
0.1
1
moisture content.
Y Vg(0.01.1.5,.001,.3502)
Value
n
a
0.013776
1.5492
0.2012
0.0019766
0.062442
0.07503.
0.35028
Chisq
y = bc(.01,.1,1,.3502)
Value
Error
Error
X
W,
0.0003499
0.39262
0.982621
Chisq
NA
N
2
0.01368
0.5428
4.7178
0.0019734
0.059962
1.7052
0.385
NA
0.962441
NA
0.0
to
0
Y.
0.0011
0.000 11
-,z
1
I'S
1
1-6
Z
ts
C
0
C.)
.S2
r7
1
Ila
1
1-8
1
'9
1
x
1
'
lu I
I
I
.
0.01
I - ,
I
0.1
moisture contem
140
I
I
.
I
.
.
I
1
0-113
eU
4
zo
0
3
3
-
2
>I.
3
1
In
q
0.
M
Q
co
0
0
9
-
van Genuchten
Brooks Corey
-1
_n
r-
-
0.( )i
0.1
1
moisture content,
y = vg(0.01,1.5,.001,.3005)
Value
Error
0.01 6809
1.6849
0.067076
0.10275
0.026421
(I
0.30003
0.0001039
8,
0.1
0.01
NA
0.98906,
NA I
i
I
z
0C
U
.2
2
.
0.016597
0.0018959
x
0.66471
0.096583
13.355
5.1152
0.51021
0.97514
NA
NA
I
Chi-sq 1
R 1
I
.
I
I
I
I
.
.
I
I
r,
I
0.001 r
.a:
U
I
of
F
0.52793
0.0001
r
1 0-5
r
I
11)
In
g
0.0018438
n
a
Chisq
y = bc(.01,.1,1,.-3005)
--- VaTue Error
Van G
-
-
- Elmok
[3
Centri
7
0
7-i
13
-i
06 r
07 r
7
7
08 r
-I n9
7
I
I
I
I
I I I I I-
.
-
. -
.
.
I
I
I
.
I w
0.1
0.1 31
moisture content,
141
1
1-1417
5
I
.
I
I
I I
.
I
.
I
I
I
I
.
I
I
-L-
I
I
.
I
.
4
w
3
0
Ci
.12
14
C
2
P
IIQ
1
C.
M
U
f=
0
0
-
e
vai
-
- Bn
-1
3e
-r-
.
I
I
I
-
.-
-
0.1
0.01
1
moisture content,
y = vg(.01,1.5,0.001,.355)
y = bc(.01,.l, 1, 3552)
Value
0.03439
Error0.003364
n
a
0
1.8032
0.0047201
0.355
0.069208
0.00054206
1.9699e-05
Chisq
0.22374
NA
Chisq
R
0.996131
NA
R
Value
- Error
ar
x
0.030052
0.65821
0.0037713
0.055299
Wb
132.74
0.1138
16.593
NA
0.99211
0.001
0
0
0
Y.
0.0001
i 0-5
-2:
'D
C
i
0-6
0
C.)
.S2
0
07
SSHC
..
L.J
so
0
V
-
.-. I
van Genucrizen
-
1 0-8
- .- BrooksCorey
[:3
Centrffuge
0
09
--
0.1
I -
.- .-1
moist= content 0
142
1-1419
5
.
.
I
I . I
I
I-I
I
4
0
0
3
C
.2
Ca
2
p
1
'E.
U
to
10
0
e
-
-1
-
van Genuchten
- Brooks
Corey
-2
0.01
0.1
1
moisture content,
I
I
y = vg(0.011,1.5,01,301)
Br
a
0 1
A
.
.2
a1I
0.001027
0.22949
0.99651
Y = W.011,11,11-30113)
0.0066
0.38MI
x
Wt,
Chisq
NA
NA I
L
error
0.0023264
0.035958
1.7186
0.22839
R
I
0.59544
NA
2-9933a5
NA
0
r
0
r
06 r
0
ZO
0
U
0
U
0.30153
R
In
!d
Vror
0.0023296
0.03664
0.20404
Chisq
04
9U
Value
0.0067962
1.3821
0.56867
08 r
r
i 0-110
x
0
r
-
r
[3
SSHC
Van Genuchten
-Brooks Corey
Centrifuge
.i n12
0A )i
0.1
moisture, contem 0
143
1
2-1636
5
I
I
.
I
I I
.
I
I
I
I
4
w
0
ci
3
0
9
'S.
M
U
Oa
2
1
0
Is
0
-
-
van Genuchten
- Brooks
I
-1
Corey
I
I
0.01
I
I
' ' I
I
I
0.1
.
1
moisture content,
y = vg(O.01 15,01,32)
0
Value
0.014573
Error
0.0017608
a
0
1.4781
0.32462
0.31993
0.046843
0.13338
0.013818
Chisq
0.351781
0.991751
I
0.001
I
NA
NA
I
I
I
y = bc(.01..1,1,0.3073)
I
r9
x
'Vb
Chisq
.I
R
I I I I
d
Z
Q
z
0
U
.2
73
i!
Error
0.014483
0.47367
0.0017151
0.043906
3.2179
0.33842
1.0071
NA
0.98MI
I
O
I
- Value
NA I
I
O
I
I
I
'i
i 04
r,
05
06
r
r
I
07
r
I
1 0-8
r
09 r
I
'SHC
(an Genuchten
>N
i 010
3rooks Corey
enfttfim
r
1I n-11
f
.
I . . . .I
0.()i
0.1
moisture content,
144
I
.
:-1
-1
-I .,,la
. I . z.
.
1
1
5
2-1637
1
.
I
I
,
I
,
I
I
I
.
I
I
I
V-T-
.
I
I
I
I
4
0
to
.2
C
3
-2
C
p
2
0
M
0
1
40
.2
0
1
I
-1
I
.
.
.
I
I
.
I
..
0.1
0.01
1
moisture content,
y = vg(0.01,1.5,.001,.3026)
(I
a
06
Chisq
y = bc(.01,.1,1,.3026)
Value
Error
0.010928
1.4607
0.26944
0.31404
0.0033498
0.071127
0.16396
0.01487
0.67881
NA
0.98251
NA
0r
1
'Vb
Chisq 1
R 1
Value
0.010847
0.45719
3.8878
0.66254
0.972361
Error0.0032728
0.067387
1.9031
NA
NA
0.01
0.001
00
0.0001
Ci
1 0-5
.2
.0
In
a
p
.a
C*
U
00
06
07
.2
1 o-8
09
0.01
0.1
moisture contem
145
1
2-1638
I
5
-- r-
1.
4
.2
Ci
3
.2
'A
C
P
'S.
M
0
2
1
to
.2
0
II
- - Brooks Corey
I
-1
.
I
I
.
I
0.01
I
I
I
.
.. . . .
0.1
1
moisture content,
y = vg(0.01,1.5,.001,.272)
y = bc(.01,.1,1,.272)
a
0
Value
0.0061047
1.4112
0.06156
0.2715
Error
0.0096753
0.095528
0.032267
0.0017373
Chisq
0.67359
NA
R
0.98051
NA
(f
0.01
.
.
I
0,
x
Wt,
Chlsq
1
.
.
I
Value
0.0051153
Error
0.0096475
0.39605
14.312
0.08651
6.8591
0.6396
NA
0.965861
NA
I
IP'
..-I
U
4
0.001
0
0.0001
0
;
.
SSHC
Van Genuchte
- Brooks Corey
I
.
.
I
7
Centrifuge
.-T
lo-.5
U
0
06
U
.2
73
Z!
07
r-
I
1 0-8
.:
0
09
I
I
.
.
I . 1. 1 ,
0.01
0.1
moisture contem
146
I
I
.
I
. I . .
1
2-1639
5
I
4
I
-- T-
'O
0.
D-
w
.2
Ci
.I
3
2
9
1
M
Q
Oa
0
0
e
van Genuchten
Brooks Corey
-1
-2
0.1
0.01
1
moisture content,
y = vg(0.01,1.5..001,.3206)
of
a
0.1
Chisq
R
Value
0.010065
1.4194
0.72627
Error
0.0026125
0.051567
0.34925
0.3222
0.69688
0.0022284
NA
0.98815,
y = bC(.01,.1,1,.3206)
value
x
Wb
Chisq
R
NA
.
I
I
I I
0.001
1 0-5
11
'O
NA
I
I
I
I
I
I
-1
I
07
0
U
U
V
>I.
0.61831
NA
-I
0
A
U
1.3502
0.69159
-1
1-1
Q
1-1
4
Error
0.0026034
0.050256
0.97939
I
1
0.0099868
0.41674
r
09
I
I
I
'I
lo-11
0.01
I
.
I . .
.
I
I
0.1
moisture content, 0
147
1
2-2225
I
5
I I
.
I
I
I
.
I
.
I
I
I
.
,
I
.
.
. I I. -
4
to
.2
3
C
.2
W
2
q
1
'S.
M
U
00
2
0
-1
-1
I
-0r-
I
I
--
I'-
.
-
. I -
I
0.01
0.1
1
moisture content,
y-
%,- . nin ni. , i, .- ,,. nni
innaj
-.
Chisq
R
0.89204
0.98236
0.01
1-1
0
4
11
11
= bC(.01,.1,1,.3309)
Error
0.00071768
0.12651
0.012727
6.6737e-05
Value
0
0.019164
1
n
1.9237
a
0.036274
0
0.323
a
0
x
Wb
Chlsq
NA
NA ..
=5
Error
0.018334
0.7659
15.187
0.0008339f i
0.07841
4.4197
0.51313
0.97795
R
NA
NA
.I
f
0.001
Cl
0.0001
0
i 0-5
U
0
U
.2
:S
M
1.
>1
x
1 o-6
07
08
09
I
I
I
I
I
I
0
1
0.1
0.01
moist=
148
content 0
Centrffuge
I
I
I
I
1
2-2226.
5
4
to
.2
3
C
.2
n
2
P
1
'E.
M
U
Oa
.2
0
-1
-2
0.01
0.1
moisture content,
1
y = vg(0.01,1.5,.001,.2861)
Value
Error
Of 0.01235
1.399
a 0.86617
0, 0.28751
Chisq 0.47278
R
0.1
E
y = bC(.01,.1,1,.2861)
0.0028043
0.06323
0.58884
0.0029354
NA
.
I
Value
0.012324
Error
0.0027928
x
0.39818
0.062431
R
. I I I I
0.001
Q
e.
i 0-5
07
I
./
NA I
.
I
I
.
I
I
-1:2
Ir
0
-M
L-
I-i
r
I-i
F
I
F
r
7S
M
>1
0.9802
0
0
0
U
.2
0.74966
NA
10
LI
In
1.1562 0.47167
Wb
Chisq
0.99077,
I
0
09
F,
1-
i 0-111
0.01
I. I...,
-M
'SHC ,
fan Genuchten
Irooks Corey
centrifuge
I
-M
I
. I . . II
0.1
moisture content,
149
1
2-2227
5
I
I
.
I
. I . . I
4
D*0
.2
3
Ci
-ICn
2
10
1
.S.
M
U
00
a
0
- - -
-1
. -
-2
0.01
t-
%wl %aI wY I
Li I
.
I
I
I
I
i I
0
.
.
I
.
,
0.1
.,
I
moisture content,
= vg(0.01,1.5,.001,.3271)
Value
Error
Of
0.014749
0.00059032
n
1.7259
0.063859
a
0.11949
0.034382
0,
0.32706
0.00012236
Y
Chisq
0.42217
NA
R
0.99192,
NA
y = bC(.01,. 1 1,3271)
09
X
'Vb
Chlsq
R
-- VaFue
0.014706
0.71812
Error
0.00059924
0.062307
8.0089
0.42232
2.2773
NA
0.98339
NA
0.01
0.001
`9
0
d
0.0001
i 0-5
U
73
0
06
U
.2
75
Cs
S>1
0
1 o-7
0
i 0-8
SSHC
Van Genuchten
Brooks Corey
0
0
Centrffuge
09
0.01
0.1
moisture content,
150
1
2-2228
I
5
I
I
.
I
I
I
I
I
.
.
I
.
I
I
I
I
I
I
I
I
I
4
M
Q
Cf
.2
'A
3
2
0
1
'S.
M
0
Da
.2
0
e
-
-1
-0r_
-
van Genuchten
- Brooks
I
I
Corey
I . . . I
I
I
0.1
0.01
1
moisture content,
y = vg(0.01,1.5,.001,.2925)
(f
n
a
es
Chisq
R
Value,
0.0095593
1.5019
0.51143
0.29107
0.85624
0.98338,
Error
0.0021178
0.095
0.39345
0.0017596
NA
NA
y = bC(.01,.1,1,.2925)
x
'4b
Chisq
.I
R
Value Error
0.0095244
0.002121
0.49983
0.09377
1.8993
0.85547
1.4338
NA -
0.96466
NA
0.01
0
1-1
4
0.0001
2
0
0
06
U
.2
LI
>11
08
i 0-10
0.01
0.1
moisture content,
151
1
2-2229
5
I
.2
W
P
I
I . I I
I
I
I
I
0
4
00
O
Ci
.
.
I
.
3
7
2
0
1
'a
M
U
do
0
0
e
-
.1
-2
I
J
van Genuchten
- Brooks
I
I
Corey
I
I I I
0.01
I
I
.
I
I . . .,
0.1
1
moisture content,
y = vg(O.01, 1.5,.001,.31)
0
Value
0.012465
Error
0.0015959
n
a
0,
1.5067
0.43734
0.30789
0.068369
0.24184.
0.0011562
Chisq
R
0.752071
0.985541_
y = bc(.01,.II..307)
NA
NA
0
Value
0.012452
Error
0.001578
x
0.50517
0.066799
W,
ChIsq
2.2607
0.74234
1.2114
NA
R
0.970521
NA -
0.001
I
0
1-1
d
13
i 0-
00
A
C.)
Lr
07
0
r
.2
F
U
z
E
>1
x
0
0
0
09
SSHC
-
Van Genuchten
-Brooks Corey
0
Centrffuge
lo-11
0.01
0.1
moisture contem
152
1
2-2230
5
.
I
4
co
11
Ici
3
.2
la
2
p
I
n
1
'S.
M
U
to
0
.2
9 Ivan Genuchten
- .= -BrooksCorey
-1
I
-2
0.01
I
I
I
0
. . I . 1
0.1
I
moisture content.
y = vg(0.01,1.5,.001,.3309)
0r
a
Bs
y = bc(.011,11,11,3309)
Value
Error
0.033799
1.517
0.017537
0.34016
0.037279
0.37127
0.024204
0.034639
7.0926
0.862451
NA
NA
Chisq 1
R 1
Value
0,
0.03499
O.OOW853
X 0.52467
0.07065
Wb 59.552
Chisq 0.52205,
10.137
NA
R
0.977581
0.001
'a
Error
NA
7
.
.
=
0.0001
I
rj
d
i O-.5
0
06
0
.2
J o-7
U
'SEM
r.
I
I
I
0
O
Van Genuchten
0
- - -Brooks
Corey
c3
08
7..
09
SSHC
--I
I
CenUtfuge
0
___ -
I
I
- __-
I
0.1
___.__-
.- ___.
L1
moisture contem 8
153
2-2232.
5
I -
i - -
v
----U
----
II
I
I
I
.
I
I
I
I -
I
.
I
I
.
.
I ,
4
.2
04
3
ci
.ICa
2
2
1
'S.
m
U
0
to
.2
- - -
-1
Lil
VJFQ
.
I
-2
0.01
wwf
I
vy
I
I I I I I
0
0.1
1
moiswm content 0
y = vg(O.01.1.5,.001,.25)
___VaTue .
ar
n
a
1
Chisq
R
0.1
0.01
1-1
_d
U
t
y = bc(.01,.1,1,.245)
Error
0.013081
1.5096
. 0. '2186
0.0029082
0.088386
0.060686
0.244
0.74951
0.0001,9375
0.985121
I
(I
I
Error
0.012761
0.0029301
0.49613
Wb
Chisq
7.3941
0.72414
R
NA
I
- -_Value
0.08243'-
3.5479.
NA
0.96881
I . I . I II
NA il
, , , , ' '!
I
.Ip
r
0.001
-i
4
6 0.0001 r
U
0
U
.2
7S
O5 F
-3
I
1O
0
1 o-7
>1
-
Lr
-
rl
1 0-8
I
SSHC
Van Genuchten
-Brooks Corey
(Igantrifima
__.
I
13 .
09
I
.
- __
I . . . .I
0.01
0.1
moist= contem
154'
-
-i
I
.
I I . . I
1
2-2233
5
V-r-
4
co
.2
0
3
00
Ci
.2
'A
C
14
2
IDOO
0
0
1
'S.
M
U
co
.2
0
o
-1
y-an Oen_pchten
BroOKS orey
0
A
-2
0.01
I I I I I
0.1
1
moisture content 0
= vg(0.01,1.5..001,.30)
Value
Error
(f
0.0072686 0.0034702
n
1.3818
0.051644
0.49604
0.24304
Ot
0.29122 0.0011704
Chisq
0.446761
NA
0.99161
NA
I
I -- .
y = bc(.01,.1,1,.2906)
1
Wb
1 Chis
I
Value
0.0071451
0.37914
Error
0.0034447
0.050094
1.963
0.92204
1
0.44223
NA
R 1
0.98331
NA
F T---IVT
I
I
I I I -
0.01
-i
w
I-I
U
:d
0.0001
01
2
zz
Iz
-I
:3/
-I
i .0-6
I-I
0
U
.2
'S
.In
>1
3SHC
08
I
/an Gen=enl
-1
3rooks C ry
I
dentrifuge - I -I
X
i 010
I
I
I
.
I
.
0.01
.
. I
0.1
moisture content,
1,55
I
I.
.
. . . . .
1
2-2234
5
7
1
1
1
1
1
1 1 1
1
I
.
.
I
I
I.
I
I
.
.
I
.,
4
30to
Q
c
3
.2
2
P
1
'S.
M
U
to
Q
0
e
-
-1
van Genuchten
-
-2
0.01
- Brooks
I
I
.
.
Corey
I
I
.
I I
I
0.1
moist=
1
contcnt 8
= 40.01,11.5,0011,28)
Value
Irror
(11 0.0088423
n
1.5142
Ot
0.071112
1
w11
Q
1-1
4
U
0
U
.2
7S
2
y = bc(.01,. 1 1,2782)
0.0028522
0.0603
0.020706
Chisq
0.27805
0.33355
8.926le-05
NA
R
0.993391
NA
0.1
LI
0.01
ff-
r
0.0001 r
i 0-5 r
06 r
0.001
1
(,
Value
().008M74
Ermr
0.0028406
x
0.48955
0.052405
11.954
3.1944
Wt,
ChIscl
R
0.29779
0.987
I
0
ssll
-
-
I
I
NA
NA
I
I
I
I I !I
-3
2
Van
- Broc
Ceni
-iz
-M
I3
0
-M
e13O
r
08 r
I
-I
[Y
2
U
n-9
%w
0.1
0.01
moist= contem
156
I
Appendix D: INEL Curve Fits
157
u(a) 30 cm
4
I
I
0
3
:20-
2
.M
P
1
0
M
Q
-1
..
-
- BrooksCcirey
-
-Campbell
van Genu(.hten
Measured
Tension
0
I
I
-2
I
--
I __
I
I
I
0.1
1
moisture content,
y = c(.21804,.II..40594)
Value
y = vg(O.29,1.6,.02,.47)
y = camp(l,.l,.40594)
Error
value
Error
W,
20.234
2.3919
x
0.2523
0.018848
,Yb
39.053
4.8546
b
13.785
0.97461
Chisci
0.083375
NA
Chisq
0.37735
NA
0.95731
NA
R2
0.92594
NA
Va(us ,
0,
n
OL
11
0.001
I
I
0.21804
0.042184
1.2967
0.013416
0.10657
0.0014499
0.40594
el
Chisci
0.057853
R2
I
.
.
.
:
0
'a1-1
U
-
0.0001
-
K (cm/s)
-
- Campbell
Boks Corey
Van Genuchten
4
2
0
05
U
9
U
06
.2
'S
Lo
>1
07
i 0-8
I
0.1
I
I
1
moismm content, 0
158
0.00IN49
NA
0.988651
I
Error
NA
u(b) 30 cm
3
I---------T-
I
I
I
I
2
.J-2
to
.2
Ii
1
0.2
C
9
>1
1.
0
0.
M
Q
-
Brooks Corey
Campbell
-
-
co
.2
van Genuchten
1
0
Measured Tension
-2
0.1
1
moisture content,
y = bc(.21,.1,1,.45422)
Value
Error
X 054916
W, 28.417
hisq
I
R
y = camp(l,. 1.45422)
Value
Error
0.032477
3.1135
040311
NA-
02561
NA
W:
b
Chisq
0.21838
NA
0.9791
NA
R2
I
.1 -
Nd
A
Q
0
U
.2
n
Cx
(I
Chisq
I
-
- IError
1.6577
0.022465
0.45422
0.26895
R2
0.039722
0.0025536
0.0024649
NA
0.97426
I
I
I
I
I
I
I
.
.
.
NA
I
- Brooks Cori
Q
1-
r (rrvs)
0.001
-
Value
12.067 079685
7.0266
0.194
0.01
I
v = v(0.21.1.5,.01,.47)
I
-
- Campbell
Van Genud
0.0001
I
1 0-5
06
07
'S
E
08
09
i 010
I
I
0.1
1
moisture content 0
159
d(a) 30 cm
3
2
00
.2
ci
1
.2
w
-
P
>1
1
0
0.
M
U
00
0
-1
-2
0.1
1
moisture content 0
I
I
y = camp(l,.1,.41295)
y = bc(.20605,.1,1,.41295)
Value
Wb
Chisq
R2
Error
0.85273
0.018229
70.821
2.1508
0.012539,
NA
0.99321
NA
I
W,
b
Chisq
Error
28.038
3.0021
VaIIIA
Value
5.858 0.35561
0.71241
NA
a
ChIsq
R2
I
0.001
1-1
I
-1- --
I
I
I
.
I
:
0.0001
0
1-1
4
i
05
=
U
1 06
L-
0
U
.2
75
LI'
PIII
x
i
A
o-7
.
.
I
n
K cm/s)
-- I
-1
- Brooks Corey
-
- Campbell
-
08
09
-
0
I
:
Van Genuchten
I
0.1
I
.
.
1
moisture content,
160
F;rrnr
0.20605 0.0041554
2.0397
0.061143
0.0096232 0.00027747
0.41295 0.00026464
0.045589 I
NA _
0.9944
NA I
a
LA
.912571
y = vg(O.22,1.5,.01,.4299)
d(b) 30 cm
3
2
D.
co
q
ci
.3
1
C
IP
>1
3
0
0.
M
0
Brooks Coiey
00
Campber..
.2
van Genuchten
Measured Tension
1
0
-2
0.1
1
moisture content,
y = bc(. 19438,.1,1,.4598)
Value
Error
x
0.56356
= q(0.24.1.5..2..49)
Value I
Error
I
0.19438 1 0.015748
y = camp(l,.1,.4598)
Va-fu-e -Trror
0.013974
W,
23.952
Y,
Chisq
55.944
0.005293,
2.1822
NA
b
Chisci
--6.2032
0.9229,
R2
0.996331
NA
R2
0.8483
3.6751
0.0852891.7552
0.0093944 0.0004772
0.4598 0.000252651
n
a
063622
NA I
0
Chlsq
NA
0.037386
0.99385
R2
NA
NA
0.01
In
0
0.001
-
-
U
d
;;
U
-
K (crrds)
Brooks Corey
- Campbell
Van Genucht
0.0001
1 0-5
C
0
U
.2
75
06
07
0
r-
1 0-8
-n
.
.
a
I
I
0.1
.
.
.
I
.
1
moiswre content, a
161
u(a) 8
cm
4
I
I
- I
I
I
0
3
00
.2
2
C
.2
'A
_
P
1
>1
a
0.
M
0
Q
do
- -
a
-1
I
0
Brooks Corey
-Campbell
van Genuchten
Measured Tension
I
-2
I
0.1
1
moisture content 0
y = bc(. 18225..1,1,.43814)
Value
-"Tr-ror
038372
19, 26.484
Chisq 0.17119
X
R2
0.96093
0.020689
2.5945
NA
NA
11
Pm y = vg(O.23,1.5,.01,.47)
Frrtr
Vilna I
y = camp(l,.1,.43814)
W,
b
Chlsq
Value
Error
17.828
1.3016
6.9436 0.25129
0.11409
NA
0.97947
NA
I
11
a
58
Chisq
R2
0.18255
1.4808
. 0020355
0.026686
0.12545
0.0033093
0.43814
0.0045193
0.068051 I
i
0.987761 I
0.01
1-1
_9
0.001
Q
4
0.0001
1 05
Ij
0
U
06
.2
73
2
>1
08
i 09,
0.1
1
moisture content,
162
NA
I
I
U(b) 8
cm
3
2
00
.2
C
.2
W
1
P
>1
a
0
0.
M
U
00
.2
1
-2
0.1
1
moisture content,
y = bc(. 18,. 1 1,.4463 1)
Value
Error
X 053909 002929
W, 24.352 2.6274
hisq
R2 1
W,, 7.2523 088418
b
6.8579 030169
Error
n
a
1.5199
0.042876
0.033581
0.0058183
0'
0,44631
0.0021862
0.34421
NA
Ghisq
O.e9167
NA
Chisq
0.71343
NA
0.93648
NA
R2
0.94686
NA
R2
0.945191
NA
0.01
I-
y = vg(. 81.5,.01,.49)
Value
y = camp(I,.1,.44631)
Value
Error
a
0.001
0.0001
05
6
la
I
I
n
I
I
.
z
r
-i
9-
-i
r
I
r
n
d
U
0
U
.2
V
>1
10-6
07
i 0-8
09
-1
K (cm/s)
- Brooks
CoreyI
F
. - Campbell
F
Van Genuchten
P
i 010
I
0.1
I
I
I
.
1
moisture conteM
163
d(a) 8
cm
3
2
04
.2
Ci
-I'A
1
P
0
0U
to
.2
1
-2
0.1
1
moisture content,
v = v(0.22.1.5..01..51)
y = bc(. 1 397,.1,1,.46448)
Value
X 0.33613
W, 22.524
Chisq 0.21622
R2
0.96604
11
y = camp(l,.1,.46448)
Value -- error
Error
0.013443
2.0674
NA
NA
W,
b
hi
R2
13.668
0
1.4142
n
6.0857 024868
a
0.46223
NA
0.9W39
NA
0
Chisq
R2
value
0.1397
1.4074
0.025617
0.46448
0.17314
Error
0.041715
0.10974
0.0038071
0.0033816
NA
0.98441
NA
0.01
In
U
d
A
0.001
0.0001
1 0-5
0
0
U
.2
SM
1.
i 0-6
07
08
09
0.1
1
moisture content,
164
d(b) 8
.
3
cm
C6
-
I
-
I
I
I
-
.
.
.
I
I
I
.
I
2
to
.2
CF
1
0
r.
P
.'A
>1
laI
0
0
0.
M
- -
U
00
-
.2
Brooks Corey
-Campbell
van Genuchten
Measured Tension
1
0
,^
I
P%
-
.
r_
0.1
1
moisture content,
= camp(i..I,.441)
v = MI -I OR - 1- - 4411
I
- Value ----- Error 11
X
'O.40873
0.012607
W.
W,
2G.081
1.5934
b
Chisq
0.096982,
R 1
0.989711
NA
= vg(O.23,1.5,.01,.53)
III
Error
of
13.24
1.1161
n
7.1098 0.25458
a
Chlsq
0.32928
NA I
R2
0.96654
NA
error
0.012197
0.059505
0.0016475
0.0012841
vetlue
Value
0.18362
1
1.5093
0.02214
0.441
NA
-0.0662761
0.99327
0.01
1-1
0
4
NA
Li
0
0.001
0.0001
A
e=
1 o-5
Ij
0r_
U
06
.2
S
V
>1
1 o-7
I
-
08
§
-
15
-
K (nVs)
I
-Brooks Corey
Campbell
Van Genuchten
09
0.1
1
moist= 6ontem
165
u(a) 145 cm
3
I
I
I
I
F---
.
.
2
Dco
.2
Ii
.2
14
1
P
0
'S.
0U
to
-
.2
-
- BrooksCorey
- Campbell
van Genuchten
1
0
Measured Tension
-2
.
-
.
I
I
0.1
I
moisture
ontents 0
y = vg(O.28,1.5,.01,.51)
--=
x
Wb
Chisq
R 1
%11.
- I.-
C(.ZbfJJU,.II,.44W)
= amp(11,1,449)
I
Value
0.46938
Error
0.031922
V,
29.354
0.036453
3.9281
NA
0.98842!
NA
.Value
Error
7.0915
1.0114
a
b
12.661
0.698
0
Chlsq
014292,
-...
R
0.001
0
K (m/s)
Brooks Corey
-
Campbefl
-
I
4
6 0.0001
Van Genuchten
'S
I
I
I
I
1.6141
0.024229
0.449
0.60748
0.98266
I
.
.
=
0
.1
1 o-5
I
1 o-6
E
E
x
1 o-7
.2
1 0-8
0.1
I
moisture content 0
166
V IVW
0.062924
0.58766
0.0094899
0.00015173
NA
NA
I
0
U
.2
R
I
0
U
=1
Chisq
0980811
0.01
Ca
n
NA
.-
Valucr I
0.26633 I
W.0
u(b) 145 cm
3
I
I
I
I
I-- _
2
00
0
Ci
1
.0
In
a
q
>1
!5
0
M
co
Q
00
.- .-_IJ_
- 9- - Brooks
Corey
-
-Campbell
van Genuchten
Measured Tension
.2
1
0
___1
-2
.
0.1
1
moisture content,
.
- .'.
- 7- .Iv = voiO.27_1.5__01__S'A
"-a 175
Error
0.015227
0.1893
r
0.041672
1.3326
a
0.017178 0.00094705
I
0
0.40444 000097431
S
Chisq
NA _
0.0023121
I
y = bc(. 1 893..1,1,.40444)
___VaTue
rror
x
0.27584
11 Wt,
Chi
y = camp(l,. 1.. 40444)
Value
rmr
0.0099292 .
W,
19.144
1.5734
- 34.801
2.2849
b
10.519
0.010426
NA
hisq
0. 1N92.
049357
NA
0.99229
NA
R2
0.972171
NA
R2
0.99939
0.01
0
ta
2
0.001
U
IV
0.0001
1 0
0
U
0
z
E!
>1
0
- Campbell
Van Genuchten
11U
d
;;I
K(cm/0
- Brooks Corey
1
.5i
O-e I
1 01 I
i 0-8
0.1
. 1
moistwe content,
167
NA
145 cm
-d(a)
3
I
I
I
I
I
"IN
I
I
.
2
to
0
ci
.2
'A
1
Q
>1
I..
M
0
0.
M
Q
-
-Brooks
Corey
-
-Campbell
00
van Genuchten
Measured Tension
.2
0
1
-2
0.1
.
1
moisture content 8
= W. 13724..1,1,.58272)
Value
0.25311
5.1503
x
W,
1
Chisq
W
0.0065054
1
0.998281
y = camp(1,1,58272)
Error
0.0029118
0.17068
Value - - -error
3.4767
0.14557
6.6515
0.085799
lyl:
b
NA 1
NA I
Chisq
0.00818381
NA
0.997841
NA
R 1
y = vg(O.Z5.1.5,.01,.51)
Value
Error
n
CE
a
Chisq
R
0.13724
0.022232
1.2651
0.03659
0.083206
0.039498
0.16109
0.58272
0.0028162
0.99963
0.1
4M
0.01
0
U
4
t.)
0_
U
.2
-
0.001
-
K (crtVs)
Brooks Corey
-
- Campbeg
Van Genuchten
0.0001
10-5
06
ff
>1
07
08
0.1
1
moistu:re conte
168
a
NA
NA
d(b) 145 cm
3
I
I
I
I
I
I
2
D_
co
.2
C
1
.2
In
14
>1
It
0
Q.
M
U
00
- -
Brooks Corey
-Campbell
-
.2
van Genuchten
-1
0
Measured T
-2
0.1
1
moisture content,
y = bc(.23377,.1,1,.57164)
Value
Error
x
0.50226
W,.
16-14
1.1431
b
0.068738
NA
Chisq
0.98197
NA
W6
Chisq
0.018884
y = vg(O.27,1.5,.01,.57)
Value
Error
y = camp(l,.1,.57164)
Value
Error
0
10.137 058275
5.8214
014717
0.031421
NA
0.991761
NA
n
a
(S
1 Chisq
I
0.01
I
0.001
Q
1-1
4
-
0.0001
0.
-
0.23377
0.005764
1.5894
0.03951
0,57164
0.003687q
0.041098
0.0036966
0.0065464
.
NA
R 1
I
I
0.999521
I
I
.
NA 11
=
Z
-M
K (cm/sec)
- Brooks Corey
- Campbell!
Van Genuchten
;;
1 0-5
U
0
U
.2
06
:5
E
>1
07
x
71-
08
i 09,
I
0.1
I
I
.
.
1
moiswre content,
169
u(a)-225 cm
3
I
I
, -
I
2
Dto
.2
Si
1
.2
CA
r2
>1
.5
0
Brooks Corey
M
U
-
-
- Campbell
bo
van Genuchten
Measured Tension
.2
1
0
I
I
r
0.1
1
moisture content,
y = bc(.22335,.1,1..46436)
Value
x
Y4
1 Chisq
I
R 1
Error
0.39026
0.016575
38.517
2.6782
0.0200161
0.993751,
I
y = camp(l,.1,.46436)
y = vg(O.28,1.5,.01,.54)
"'7=7
Value
--W,
1
NA 11
NA
b
Chisq
R
20.912
1.8222
8.4134
0.4362
0.12874
0.98297
Error
0.014222
0.060319
1.4823
0.015132 0.00082392
0.46436 0.0013885
. 0.22335
9
a
NA
NA
0
-I:-%'111b4
110%
0.99M
NA
R2
0.01
'a
I
0
0.001
-
0
4
-
.11.
V.Uvu I c
I
I
:
IK (nVs)
Brooks Corey
-
0
- Campbell
Van Genuchten
;;
0.0001
0
05
0
U
P
75
LI
>1
x
i 0-6
1 o-7
-2
08
I
0.1
I
I
I
I
1
moisture content, 0
170
I
3
u(b) 225 cm
e
S
I
I
I
I
I
I
I
.
0
2
BO00
.2
CF
1
.2
Cn
C
9
>1
I;
0
0.
M
-
U
00
-
- BrooksCorey
-
- Campbell
van Genuchten
.2
-1
0
Measured Tension
I
-2
I
I
a ,
0.1
1
moisture content,
I
y = bc(.30044,.1,1,.45122)
x
Wt,
ChIsq
R2
Value
0.55632
Error
0.045027,
76.218
8.6571
b
11.51
0.022848
NA
Chisq
0.15002,
NA
NA
R2
0.947221
NA
0.97449
7
A
AR)
Error
0.30044 0.0088312
1.7724
0.096068
0.0068ST2 0.00025951
0.45122 1 00006644
Value
7515-5 --- E-rror
37.44 3.6554
W,
v = v(n
-
I
y = camp(l,.I,.45122)
r
n
a
081923
0
Chisq
0.0058,46-1 I
Z:f
0.99794
0.01
I
.0
0.001
-
U
d
;;I
U
a
0
U
- --
0.0001
I
K (cm/sec)
Brooks & Corey
I
I
I
I
:
0
-3
-Campbell
Van Genuchten
1 o-5
i 0-6
.2
Lo
>1
x
07
I
08
-3
09
0.1
1
moisture content,
171
NA
---
NA
_d(a) 225 cm
3
I
I
I
I
I
I
I
-
I
I
2
W
q
9i
1
.0
'A
C
S
0
I"
0.
W
U
-
to
-
-
.2
- BrooksCorey
-Campbell
1
0
van Genuchten
Measured Tension
I
-2
.
"%
0.1
.
1
moisture content,
y = earnp(1,1,45246)
P1,11"__
value I Error
14.013 052298
W.
5.7495 1 010683
b
NA
Chisq
0.016754 I
= bc(. 18619,.1,1,.45246)
- __
Value
0.49396 1
_x
lyt,
Chisq
Error
0.025228
1.7676
20.28
0.12286
0.96723
NT
0.99553
NA
I
NA
I
y = vg(O.21,1.5,.01,.55)
value
n
a
Chlsq
82
0.18619
1.6023
0.027161
0.45246
0.013428
0.99642
0.01
W2
U
1-1
4
z
0.001
0.0001
05
0
U
.2
i 0-6
1!
>1
1 o-7
08
09
0.1
1
moisture content,
172
Error
0.008491
0.070581
0.0030562
0.0048208
NA
NA
d(b) 225 cm
3
E
2
.2
Si
.2
la
1
S
0
S0
Brooks Corey
-Campbell
0
00
.2
van Genuchten
0
Measuredtens!
-2
0.1
1
moismim ontent 0
y = bc(. 18229,.1,1,.46015)
Value
Error
x
0.45317
0.014177
Wb
13.595
0,047858
0.98745
0.86093
Chisq
82
NA
I
I
y = camp(1,1,46015)
II-_
VdIUU I
_rrur
a
Value
0.18229
1.514
0.048154
11 ('hi-en
-. .. ,
- 0.46015
n ni 5nw;
I
8.1387
W.
b
Chisq
. e
n
6.1479
0.016133 7-NA0.99577
NA
NA
y = vg(O.21,1.5..01,.56)
0.99683
I
0.1
I
I
'a
0
1-1
Nd
0.01
F,
0.001
S_
I
___b
-
0
K(m/sec)
- Brooks Core
-
- Campbell
Van Genuchtei
2
00
0.0001
0
1 0-5
:3
Fl
>1
1 o-6
U
.2
ff
1 o-7
2-
30
A9.
08
0.1
1
moislum contem
173
Error
0.0087279
0.062945
0.009288
0.010718
NA
NA
Appendix E: Las Cruces Curve Fits
174
Core 334
4.5,
I
I
I
.
I
.
I
I
I
I
I
I
I
I
I
.
I
I
I
I
I
. 4
3.5
M
q
CF
3
.2
In
q
2.5'
2
'E.
M
U
O&
-
1.5
0
1
0.5
-
- BrooksCorey
.- Campbell
E)
Van Genuchten
I
I
.
.
0
0.01
I
I
,
I
0.1
1
moisture content,
y = bc(.01,.1.1,.337)
Value
0r
0.054744
Error
0.0021027
x
0.46684
0.04787
wo
Chisq
3.6048
0.33426
0.9723
0.98334
NA
NA
R2
y = vg(0.01,1.5,.001,.337)
Value
Error
y = camp(l,.1,337)
VaTue --- Error
or
Wa 095922 053316
b
4.7307 05046
Chisq
1.5547,
NA
R2
0.871151
NA
a
Chisq
R2
0.055267
1.4878
0.23637
0.37649
0.9688
04
0
0
100
Q
0
1
U.,
0.01
U
0
0
U
.2
F
0.0001
Ir
r
06
It
K (cm/d)
Brooks Corey
Campbell
Van Genuchten
F
r
r
08
4
I
0
I
Lr
I 10
. . . ... ...
0.1
1
...
. . . .-..-- - - --
10
100
1000
Capillary Tension, W (cm)
175
1
o4
05
0.0021032
0.055218
0.070537
NA
NA
11
i
Core 425
4.5
I
I
I
I . . 11
I
I
I
I
4
000
3.5
.2
ci
3
0
.'A
C
P
'O
2.5
. 'i
2
M
U
00
-
1.5
.2
-
- BrooksCoi
-
Campbell
I
1
E)
Van Genuchten I
0
==A
I
0.5
I
I
I
. . . I I
0.01
0.1
1
moisture content,
y = vg(0.01,1.5,.001,.315)
Value
Error
y = bc(.01,.1,1..315)
Value
6
x
Wo
Chisq
R2
y = camp(l, 1,31 )
Error
0.040695
Value
0.016432
0.2803
0.057792
7.8747
0.4M7
2.2195
NA
0.96015
Error
(f
We 5.6952 1.2934
h
5.1302 0.31759
NA
Chisq
0.56901
NA
n
a
Chisq
R2
0.95255
NA
R2
0.04944
0.012441
1.3337
0.084432
0.44408
0.066577
0.024302
NA
0.96296
NA 'I
04
i
100
1-10
4
.:;
1
U
0
U
.2
ii
>I.
x
0.01
0
K (cm/d)
- Brooks Corey
- Campbell
Van Genuchten
0.0001
1 o-6
0.1
1
10
100
Capillary Tension, v
176
1000
(cm)
1 04
05
Core 534
4.5
4
0
3.5
0
ci
3
.2
(A
r.
P
2.5
0
2
M
U
-
0
1.5
-
- Brc
(D
I Val
-
Cai
1
I
0.5
0.01
I
I
I I I I I
I
I
I
.
.
I
0.1
. '
1
moisture content,
Value
WO
Value
Error
6.0231
1.6774
Error
X 023908 001848
6.0231
1.6774
0.85891
NA
F32 0.93316
NA
hisq
y = vg(0.01,1.5..001..305)
Value
Error
y = camp(l,.1,.305)
y = b(O,. 1 1,305)
Y
b
Chisq
R2
4.1826 0.32313
0.85891.
NA
0.93316
NA
(f
n
a
Chisq
0.0084438
1.2776
0.11054
0.93089
R2
0.927561
04
100
d
1
e
.!I.
U
0
0
U
0.01
0
.2
1!
>11
K (cm/d)
- Brooks Corey
0.0001
- Campbell
Van Genuchten
ia n-6
I
I I IIM I
.
I I . Ifill
%.F
0. 1
1
1 10
100
Capillary Tension,
177
-1000
(cm)
1 04
1 0-
0.031398
0.10189
0.05459
NA
NA I
Core 721
5
I
I
I
-
4
--
I
I
-
- BrooksCorey
- Campbell
00
.2
E)
ci
I
Van Genuchten
3
.0
rP
t,
.9
2
'E.
aU
00
0
0
1
C
0
I
0
-
I
- --
I
I
.
0.1
1
moisture content,
y = bc(.01,.1,1,.322)
Value
Of
0.044061
0.074636
x
0.1951
0.11501
WO
Chisq
R2
9.5726
1.7294
I
Error
W,
b
Chisq
4.7785
NA
0.87329
y = vg(0.01,1.5,.001,.322)
y = camp(l,.1,.322)
Value
Error
R2
Value
8.1023 2.7193
6.8135 0.69913
1.7533.
NA
0.87153
NA
0.077206
0.041637 .
a
ChIsq
1.2922
0.055128
1.7154
0.15504
0.028947
NA .
0.87431
W
NA
04
100
0
U
0
d
A
1
r
U
v
0
U
.2
i!
0.01
r
0.0001 r
I
0
-
K (cm/d)
- Brooks
Corey
- Campbell
Van Genuchten
n-6
0.I
1
10
100
1000
Capillary Tension, 4f (cm)
178
Error
0
1 04
1 05-
NA
Core 849
4.5
I
I
I
I
I
I I I 11
I
I
- I
I
I
I I
I
I
I
4
3.5
ea
.2
C
.2
3
P
2.5
'S.
M
2
0
1.5
CA
C
Q
00
-
-
1
- Brooks Corej
- Campbell
E)
6M
I
0.5
Van Genuchl
I
.
. . I . . I
0.01
--
t-
I
I .
0.1
I
moisture content,
y = vg(0.01,1.5,.001..332)
Value
Error
y = bc(.01,.1,1,.332)
Value
Error
0
0.059871
0.0056796
x
0.3638
0.049691
IYA
4.4293
1.3401
Chisq
11 .-.
W
0.38786
NA
0.966881
NA I
y = camp(l,.1,.332)
Value
Error
W,
b
1.8623
I
ef,
n
a
Chlsq
074387
5.3017 044809
Chisq
0.92445
NA
R2
0.92105
NA
R2
0.061522
1.3896
0.18
0.41614
0.96"6
0.0054165
0.057438
0.059009
NA
-
04
i
0
.0
100
K (cm/d)
- Brooks Corey
0
- Campbell
d
.t:
Van Genuc
0
1
U
0
U
0.01
.2
75
E.C*
x
0.0001
06
0.1
1
10
100
1000
CapiHary Tension, ili (cm)
179
1 o4
05
NA
I
Appendix F: Maddock Curve Fits
180
Plot I
5 cm)
2.8
I
I
2.6
00do
2.4
CF
.Q
14
Q
2.2
.
I
Brooks Corey
Campbell
0
.2
r-
I
van Genuchten
.
2
1.8
M
U
00
1.6
.2
1.4
0
1.2
0.1
1
moistwe content 0
= bc(.0 1. 1, 1,41 8)
Value
Error0
0.10464
0.20278
wt.
Chisq
R
12.175
0.022509
0.99415
0.091661
0.09427
2.3245
NA
NA
y = camp(l, 1,41 8)
-- VaTu-s-- Error
W, ,
b
Chisq
R
10.695
1.0584
7.5128 027944
0.0237161
0.993831
181
NA
NA
-. 0f,
a
CE
Chlsq
R
I
- -
,.
= gkv. 1 10,. 1,41 0)
Value
Error
0.19843
0.018302
1.4556
0.041327
0.088717
0.0043972
0.012644
0.99672
NA
NA
Plot ( 530 cm) - -
2.8
2.6
1_
Oa
2.4
CF
.!2
'A
2.2
I I
II
I
-
-
-Brooks Corey
-Campbell
E)
.2
van Genuchten
2
_P
e.,
1.8
9.
Q
1.6
91
1.4
1.2
0.1
I
moisture content,
y= bc(.1..1,1,.418)
Value
Error
0r
0.083099
0.034478
A.
0.3346
0.075458
Ivb
12.025
1.7981
ChIsq
R
0.01794
0.99534
NA
NA
y=camp(l,.l,.418) .
Value
W.
b
ChIsq
R 1
9.8333
Irror
0.98965
4.5437 0.16694
0.023147
NA
0.993981
NA
182
I
y = vg(O.1, 1.5,01,41 8)
Value
Error
of
0.1375
0.014178
n
1.6024
0.099843
0.044432
0.0054616
CNsq
R
0.01856
NA
0.99518
NA
Plot
30-45 cm)
2.8
2.6
3do
2.4
9i
2.2
-
C
P
- BrooksCorey
Campbell
e
0
.0
W
-
-
van Genuchten
2
1.8
'E.
M
U
W
1.6
.2
1.4
1.2
0.1
1
moisture content,
= bc(. 5.1,1..399)
Value
Error
01
0.11662
0.012098
x
0.43695
0.051821
ly
12.639
1.17
Chisq
8
0.0073288
0.9981
NA
NA
y = camp(l,.I,.399)
Value
W.
b
ChIsq
R
8.6163
I
Error
0.90292
4.6785 0.17122
0.0229681
NA
0.99403
NA
183
y = vg(0.1,1.5,.01,.399)
Error
Value
n
a
Chisq
0.14322
0.0031597
1.6645
0.031729
0.001869
NA
NA
0.047726
0.0018336
0.99952
Plot
45-61 cm)
2.8
I
2.6
-
0_
04
2.4
-
.2
-Campbell
E)
2.2
ci
.2
In
-BrooksCorey
van Genuchten
C
P
2
>1
I.
M
1.8
M.
M
U
M
1.6
.2
1.4
1.2
0.1
1
moisture contents 0
y = c(.1,.1,1,.39)
Value
1
x
0.11771
0.0014528
1.0399
0.06513
W,
14.98
1.1982
Chlsq
R
0.0091718
0.99762
NA
NA
y = vg(0.1,1.5,.01,.39)
Value
Error
y = camp(l,.l,.39)
Error
Value
W.
error
5.5165
1.8382
3.5225
0.3582
of
0.11979
0.00065566
n
2.2131
0.043674
0.16419 .
NA
a
0.051811,
0.0023082
RI 0 .95648
NA
ChIsq
R
0.00327831
O.M151
b
hlsq
U
184
-
NA
NA
Plot
2.8
I
I
I
I
61-91 cm)
I
I I I 11
I
I
I I I I I
.
.
.
2.6
D_
2.4
04
q
2.2
Ci
.20
2
2
1.8
'F6
M
Q
04
-
1.6
.2
-BrooksCorey
-
1.4
-Campbell
E)
van Genuchte
I I I . . .I
1.2
0.01
X
.
0.1
.
.
.
1
moisture content,
y = bc(.05,.1,1,.367)
Value
el
x
IVb
Ch1sq
R
Error
I
I
0.069535 0.00011857
2.0509
0.09MS
19.765
1.4192
0.012861
0.99666
NA
NA
-
___
J
M
y = camp(l,.1,.67)
Value
Error
W.
b
Chlsq
R
3.1269
2.317
1.8798
0.49086
0.8634
0.36614
NA
NA
185
y = vg(O.05,Z.01,.367)
- Value0r
0.069718
n
3.3949
0.039038
Chisq I
R 1
0.01 3475
0.9965
, Error
7.66799-05
0.11551
0.0025756
NA
NA
Plot 1 91-122 cm.)
2.8
2.6
to
2.4
CF
2.2
.2
.2
'A
a
2
2
1.8
M
fj
00
1.6
.2
1.4
1.2
0.1
1
moisture ontents 0
v a W I.. 1.1-406)
Value
0
A.
sq
R
0.12449 0.00036918
1.4714
0.075401
15.72
1.2926
0.013373
0.99653
y = vg(0.1,1.5,.I,.406)
y = amp(1,1.406)
Error
NA
NA
Value ----
0
Wg 82134
4.8514
b
2.8173, 058064
ChIsq
0.63333
0.85056
n
a
NA
NA
Chlsq
M
186
M
Ermr
Value
0.12507 0.00044204
2.7622
0.15592
0.045622 0.0059946
0.042843
NA
-U.UtRM
-----
NA
I
Plot 1 (1 22-152 cm)
2.8
2.6
2.4
.2
2.2
Ci
-LI
cm
_
N
2
2:1
1
Q.
0Q
1.8
to
1.6
.2
1.4
1.2
0.1
1
moisture content,
v = bc(A.A.I..4361
Value
8
1.7706
12.336
2.5197
i
WI'
Chisq 1
0 1
-
Error
0.00022898
0.18029
0.1076
x
y-
0.071614
NA
A
no I nc.
%,..
II.rI
I".
I
-7=59
W.
b
ChIsq
R
5.8964
y = vg(0.1,1.5,.01,.436)
I
Value
Error
AM
-
_r
r_Or_
2.4743 0.70733
0.81728
NA
0.75908
NA
I
187
a
5.3325
n
a
I
ChIsq 1
R
0.10768
2.912
0.06813
0.091922
0.97588
0.0002155
0.22106
0.015129
NA
NA
Plot 2 (0-1 5 cm)
2.8
2.6
-
2.4
co
-
-
- BrooksCorey
-Campbell
.2
E)
van Genuchten
2.2
ci
.2
14
C
.2
2
1.
M
1.8
C6
M
Q
*a
.2
1.6
1.4
I
1
I
I
I
..
a-
0.1
I
moisture content,
= bc(.2,.1,1,.408)
-_
Value
EfMr
(f
0.16921
0.017608
X
0.37822
0.064739
W,
13.767
1.5288
Chisq
R
0.01,159
0,99699
NA
y - camp(l..l..408)
-- Va.u-e - Error
14.
b
CNsq
R
9.4994
1.0539
6.667 026691
0.027421,
0.992861
188
NA
NA
y = vg(0.1,1.5,.01,.408)
Value
0r
0.20196
n
CZ
1.6439
0.040619
- Chis 1
R I
00042302
O.9M
Error
0.0
0.0
0.0021 "
N
NA
Plot 2 15-30 cm)
2.8
2.6
>
Oa
2.4
C
2.2
-
.
I
van Genuchten
2
I_'0
.a
.V
Campbell
E)
.2
W
I
-Brooks Corey
-
.2
w
fj
-
I
I
I
I
1.8
I
1,
1.6
1.4
I
1.2
I
I
0.1
1
moisture conten
y = bc(. 15,.1.1,.391)
x
4b
ChIsq
R
0
y = camp(1,1,391)
Value
Error
0.11447
0.013829
W.
11.176
0.94873
n
0.047972
b
4.7752
0.15471
a
0.4018
14.695
0.0063296
0.99836
1.1312
NA
NA
Valui- -- Sr-ror
y = vg(0.1,1.5,.01,.391)
Value
Error
0f
0.15481
0.001456
ChIsq
R
0.018048
NA
0.99531
NA
189
1.7328
0.016889
0.036288 000059407
Chlsq
0.00048591
NA
R
0.99987
NA
Plot 2 30-45 cm)
2.8
I
I
I
I
.
I
I
.
2.6
DMD
2.4
C
P
Campbell
E)
2.2
.2
'A
- BrooksCorey
-
.2
Ci
-
van Genuchten
2
>1
141
0.
M
1.8
.2
1.6
U
00
1.4
I
1.2
I
.
0.1
1
moisture content 0
I
= C(.15,1,1,367)
Value
0
Chis7q
i
I
0.10937
0.42027
0.017563
16.144
1.604
0.011838 I
-u.vvbU;j
-----
Y -
Error
0.06856
NA
NA
i
W.
b
Chisq
R
I
I
12C'7%
y = v9(O-I0, 11.5,01,367)
Value
Error
-- =Us
.. " I
- Error
12.47
1.1499
or
4.6204
0.16921
NA
NA
a
0.023
0.9W2
M
190
Chlsq
R
0.14865
0.0015509
1.8019
0.022325
0.032448, 0.00061347
0.000790031
0.9998 I
--
NA
NA
Plot 2 45-61 cm)
2.8
2.6
2.4
to
.2
2.2
Ci
.0
'A
a
Ip
2
.a
1.8
M
Ij
to
1.6
.2
1.4
1.2
0.1
1
moisture content,
y = bc(.l,.1,1,.363)
Value
Error
0,
0.096538
0.0019274
x
0.93136
0.059487
lyb
14.917
ChIsq
R
0.0081593
0.99788
1.1451
NA
NA I
y = camp(l,.1,363)
Value
Error
T,
b
hisq
RI
6.7066 1.7773
3.1787 0.27232
0.11949
0 .96852
191
NA
NA
y = vg(O.1, 1.5,.01,.363)
Value
or
Error
0.099873 0.00082777
n
2.1251
0.040586
a
0.049786,
0.0020914
Chisq
R
0.0029016
0.99925
NA
NA
Plot 2 61-91 cm)
2.8
I
I
I
I I I I I I
I
I
I. I
I I I I.
2.6
2.4
Oa
.2
r-
.q
2.2
'A
a
2
2
1.8
.a
M
Q
1.6
fto
0
-
-
- BrooksCorey
-
Campbell
1.4
E)
II
I
03
r
van Genuchten
I
I
I
I
0.01
I
0.1
1
moisture content,
I
y = bc(.05,.1,1,.371)
Value
Error
0.083227 0.00020785
'I't,
Chisq
H
1.8406
0.10354
18.804
1.5893
0.016827
NA
U.9V563
NA
- -----
y = camp(1,1,371)
Value
Error
y = vg(O.01.1.5,01,371)
I
--- Vafue
Error
14.
of
0.083576
n
32581
0.037864
Chisq
R
0.032599
0.991 51
10.57
b
2.0341
h1sq 0.51682,
R I SW571
192
5.5998
04102
N
N
0.00017294
0.17336
0.0038957
NA
NA
Plot 2 91-122 cm)
2.8
I
.
I
.
I . . 11
I
I
I I I . 1,
2.6
2.4
w
.2
2.2
Si
.0
w
P
2
>1
M
1.8
Ca.
M
ej
to
1.6
.2
- -
BrooksCorey
-
Campbell
1.4
e
van Genuchten
-f f3
I r
I
I
I
I
0.01
I
0.1
1
moisture content a
= bc(.05,. 1. 1,394)
Value
ar
x
Error
0.088395 0.00024335
1.9485
'Yb
16.586
Chisq
R
0.039952
0.98959
0.18001
2.3783
NA
NA I
y = amp(l,.I,.394)
Value
Error
W,,
b
Chisq
R
8.5554
5.7935
2.1139 0.50832
3.66008,
NA
0.8111
NA
193
y = vg(0.01,1.5,.01,.394)
Value
Error
or
0.088617
n
3.2814
a
Chisq
R
0.046102,
0.054075
0.98588
0.00019362
0.24742
0.006995
NA
NA
Plot 2 122-152 cm)
2.8
2.6
2.4
00
0
2.2
Ci
.0
In
rp
2
S
1.8
M
U
04
1.6
q
1.4
1.2
0.1
I
moisture content,
..
f
I.-
.
.
.
Value
0
0.1268
1.498
Wt.
Chisq
R
. -1
11
%,;k.
1'. II I..%Qv)
11.351
0.023536
0.99388
1.
1 1
V = CaMOO.J..456)
Error
Value
0.00033508
0.10417
W
3.2078
2.8694
b
3.2765
0.77755
1.4892
Chlsq
NA
R
NA
11
0.
0.81464
I
194
y = vg(O.I.1.5,.01,.456)
Error
NA
or
(X
Chisq
R
--- Taue
Error
0.12699
0.00035128
2.6059
0.0757911
0.033496
0.99128
0.13536
0.011579
NA
NA
Appendix G: Sevilleta Curve Fits
195
30.5 cm
1
I
0
I
I
I
.
I
I
I
I
I
-
I
I
.
I
I
-
- Brooks Corey
!DIO
- Campbell
0
ri
0
,n
G
van Genuchten
C
P
>1
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M
U
CLO
0
A
I
I
I
I
I
I
. I I . .I
0.01
I . . . . .
0.1
1
moisture content,
= nIO (I I - Ml
y = bc(.01,.1,1,.33)
--- VaFue
61
x
WA
Chisq
R
Error.
0.04297
.-I7aue
0.011471
21.916
b
1.0442
0.077813
0.32794
23.522
1.892
Chisq
0.0169551
NA
0.011367
0.9909
NA
NA
R
0.98641
NA
0.0010686
n
a
3.8915
0.028241
0.19589
0.0012086
8.
Chisq
0.34767
0.00076006
0.0072733
NA
R
.
I
R
0.055984
1.7972
1.5349
0.01
w
Error
0r
Error
4":
0.001
0
K (cm/S)
-
- Campbell
-
-
I
I
.
I
I
I
I
0.99939,
NA
I
I
I
I
I
I
I
I -
I
I
I
.
I
I
.
I
Brooks Core)
U
Van Genucht
4
0.0001
U
-
0
U
.2
05
75
M
>1
i 0-6
07
I
- I ---
.
I
I
0.01
I
I
0.1
moisture content, E)
196
qql
-- Varue
y = carnp(1,1,33)
1
61.0 cm
1
I
0
I
I
I
.
I I
I
I
-
I
I
.
. ' ' I
- Brooks Corey
- Campbell
CIO
P
ci
e
van Genuchten
-
.2
14
q
M
U
co
11
I
.
- 1 11 . I I I
-.-
I
.
I
I I I I
1
0.01
0.1
1
moisture content,
I
.
.
- .
.
-
-
= DC(.U1,.1,1,.J15)
Value
Error
G
x
Chisq.
R
0.050779
1.6976
21.086
0.0080611
I
0.0047932
0.24198
1.4945
0.9941
IV,,
b
18,669
- Chisq-'
0.026485
0.98049
II
NA
NA
y = vg(O.01 1.5,001,318)
Value
Error
y = camp(l,.1,.318)
Value -,Tr-ror
2.0724
0.10151
NA
NA
1.1321
R I
0r
0.056102
0.0020807
n
3.4911
0.37464
a
0,;
0.036458 0.0054763
0.35928 ' 0.036312
Chisq
0.0035467
0.997411
n i
-11
I
0.01
U
a0
I
0
-
U
d
I
E
-
-
I
I
. . . I I
I
I
I
I
T- I
I
I :
.
.
.
.
.
K (cm/s)
Brooks Core)
- Campbell
Van Genucht
0.001
0.0001
=-
U
.2
73
M
U.
>1
1 o-5
=-
1 o-6
=0
0
1 07
L
I
I
I
I
.
0.01
.
I
I
0.1
moisture content,
197
I
.
1
NA II.
NA
91.5
I
10
I
I
.
F
CM
I II
-i
-
-
I
-2
-
Campbell
e
I
I I
- BrooksCorey
to
I
van Genuchten
ci
0
.11,
P
>1
.:i
'E.
U
M
0
-1 I
I
0.1
0.01
1
moisture content,
y = bc(.01,.1,1,.325)
Value
y = vg(0.01,1.5,.001,.325)
Error
(I
0.061856
0.0074636
).
1.8389
0.44996
VP
22.467
2.445
Chisq
R
0.021005
0.98"
I
0.1
0.01
I
U
-
U
a0
U
.U
75
.1
1.
;-I
:z
Value
I
I
19.826
2.53
'v.
b I 1.1759 0.12794
0.03794
Chlsq
NA
NA
0.97166
NA
NA
d
= camp(l,.1,.325)
Value
Error
-
I
I
I
I I I I I
I
F% rrus)
0
0.068606
0.00098292
n
4.5054
0.028012
0.32458
0.48832
0.0022145
a
9
I
0.0078751
NA
NA
0.0051939
Chlsq
R
I
Error
0.99617
I
I
I I
.
I
I
- Brooks Corey
- Campbell
Van Genuchten
0.001
0.0001
05
1 o-6
07 I
I
IIIIIIII
0.01
0.1
moisture content,
198
1
I
I
i
THESIS PROCESSING SLIP
FIXED FIELD:
ill.
-
index
o,-COPIES:
CrchiDves
Aero
Lindgren
Music
TITLE VARIES:
Dewey
name
biblio
-
:E:nDg
Hum
Rotch
Science
R
No.
NAME VARIES: ),[-
IMPRINT:
(COPYRIGHT)
No-COLLATION:
/
o-ADD. DEGREE:
16y"
/ d
! &
o, DE PT.,
SUPERVISORS:
NOTES:
cat'r:
wDEPT:
-C,
wYEAR:
P-NAME
-
Ef
7`77
-775
PDEGREE:
)o -5c, r n o
date:
e
, 11
M , 5,