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This is to certify that the
dissertation entitled
Use of Scaling Theory in Predictive Methods to
Efficiently Construct Two- and Three—Phase
Constitutive Relationships

presented by

Herr Soeryantono

has been accepted towards fulﬁllment
of the requirements for

Ph . D . degree in Civil Engineering

 

 

Date April, 1995

MSU is an Afﬁrmative Action/Equal Opportunity Institution 0-12771

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Use of Scaling Theory in Predictive Methods to Efﬁciently
Construct Two- and Three-Phase Constitutive Relationships

by

Herr Soeryantono

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

1995

Roger B. Wallace

 

ABSTRACT

Use of Scaling Theory in Predictive Methods to Eﬂ‘iciently Construct Two- and
Three-Phase Constitutive Relationships

by

Herr Soeryantono

Scaling theory (Parker et al., 1987) combined with Leverette’s (1941) predictive
method permits prediction of the relative permeability-saturation-pressure (kt-S-P)
relationship of any two- or three-phase system from P-S data of a particular two-phase
system and interfacial tension data. Implementation of the theory is studied by comparing
the two- and three-phase kr-S-P curves constructed with Leverette’s predictive method
with and without the use of scaling theory. The observations include predictions which

are based on air-water or air-oil P-S data using two soils: uniform graded Ottawa sand

and loamy sand Metea soil.

Comparison of the kr-S—P curves obtained with and without the scaling theory

showed that, at least for the investigated cases, the use of air-oil system as the basis of
prediction is only suggested for a well-graded soil. When involving a uniform grain type

soil the use of air-water system as the basis of prediction is preferable.

The required P-S data were measured in a pressure cell using the pressure
equilibrium method. This method reverses the conventional technique (Klute, 1986) by
imposing rapid ﬁnite desaturation and subsequently allowing hydrostatic equilibrium
under a zero ﬂux boundary conditions. Theoretical veriﬁcation showed that this method
is more efﬁcient than the conventional technique. Laboratory experiments confirmed the
theoretical result and showed that the P-S relation obtained from the data, statistically,

indistinguishable from that obtained by conventional method.

To Ibu, Christina, Titon and Nino

iv

ACKNOWLEDGMENT

I gratefully acknowledge the contributions of those who have made this dissertation
possible. My deepest gratitude goes to Dr. Roger B. Wallace, Associate Professor,
Department of Civil and Environmental Engineering, Michigan State University, my
dissertation director and advisor during my course work. I appreciate his fatherly advice,
patience, and continual encouragement throughout my entire graduate program and his
stamina in reviewing a great many draﬁs. He was instrumental to the successful
completion of this dissertation.

I would also like to thank Dr. Thomas C. Voice, Associate Professor Department of
Civil and Environmental Engineering; Dr. David C. Wiggert, Professor, Department of
Civil and Environmental Engineering; and Dr. Raymond J. Kunze, Professor, Department
of Crops and Soil Science, Michigan State University for serving in my dissertation
committee and for their extremely valuable comments in this study.

For the further development of pressure saturation measurement apparatus which
was essential in this study I owe Dr. Arthur T. Corey, Profesor Emeritus, Department of
Agricultural Engineering, Colorado State University. Dr. Mokma, Professor, Department

of Crop and Soil Sciences helped to select the soil types I used.

Research funding was obtained from several sources. I wish to thank the Second

Higher Education Development Project, Ministry of Education and Culture Republic of
Indonesia that provided support for the ﬁrst three years of my doctoral program. This

research could not have been completed without support from the National Institute for '

Environmental Health Sciences (Grant ESO4911) and the Michigan State University

Institute for Environmental Toxicology.

Ruth Dukesbury and Kahlil Rowter helped in proofreading. For that I am very
grateful. To Mike Annable, Lizzette Chevalier, Reza Rakhshandehroo, Munjed Maraqa,

Michel Hsu, Yao J .1, Linda Steinmann and Brenda Minott, my colleagues in the
Department of Civil and Environmental Engineering, Michigan State University, I would

like to express my deep appreciation to for all their support and encouragement

Finally, I want to thank my wife and children: Christina, Titon and Nino for their

love, care, support and understanding.

vi

TABLE OF CONTENT

 

 

 

 

 

 

 

 

 

LIST OF TABLES ....................................................................................................................................... ix
LIST OF FIGURES ....................................................................................................................................... x
LIST OF SYMBOLS - - __ - ..................................................................................................... xii
CHAPTER I: INTRODUCTION ................................................................................................................. 1
1.1. BACKGROUND .............. .................................................................. I
1.2. GENERAL OBJECTIVES, APPROACHES AND PRESENTATION ORGANIZATION ............................................. 7
1.3. DEFINITIONS AND GENERAL OVERVIEW. - ................................................ 9
1.3.1. Multiphase System ................................................................................................... 9
1.3.2. Multiphase Flow Equation and the Predictive Method ................................................................... 11
1.3.3. Methods to Scale the Pressure-Saturation Relationship .................................................................. 14
1.3.3.1. General Formulation of the Pressure-Saturation Scaling ........................................................................ 14
1.3.3.2. Leverette (1941) .1 Function . ........................................................ 18
1.3.3.3. Similar Media Concept by Miller and Miller (1956) ............................. 19
1.3.3.4. Brooks and Corey (1964) Pressure-Saturation Function as A Scaling Method ...................................... 21
1.3.3.5. Lenhard and Parker (1987) Scaling Theory. ............................................ 22
1.3.3.6. The Issue of Contact Angle. ....................................................................................... 23
1.3.3.7. Dependency of Scaling Method on Soil Texture .- ....... 24

 

CHAPTER II: THE ACCEPTABILITY OF THE PRESSURE EQUILIBRIUM METHOD

 

 

TO MEASURE DRAINAGE PATH OF SOIL PRESSURE-SATURATION ........... 25

2.1. BACKGROUND ....................................... - -. - - ............................ 25

2.2. MEASUREMENT METHOD AND MATERIAL .............................................................................................. 29

2.3 . RESULT AND DISCUSSION ........................................................................................................................ 36

2.3.1. Pressure-Saturation Measurement Simulation... -. ......................................... 36

2.3.2. Hysteretic Issue in the P-S Measurement Employing the Pressure Equilibrium Method ............... 41
2.3.3. Pressure-Saturation Measurement employing the Pressure Equilibrium and the Saturation

Equilibrium methods. ...................................................................................................................... 44

2.4. SUMMARY AND CONCLUSION .................................................................................................................. 55

CHAPTER III: THE ACCEPT ABILITY OF SCALING THEORY IMPLEMENTATION IN
PREDICT ING THREE-PHASE CONSTITUTIVE EQUATIONS FROM

 

TWO-PHASE PRESSURE-SATURATION DATA. .................................................... 58
3.1. BACKGROUND- _- _ .................................................................................................... 58
3 .2. THEORETICAL BACKGROUND. ................................................................................................................. 62
3 .3. MATERIAL AND METHOD. ....................................................................................................................... 67

vii

 

 

3.4.3.111

3.5. Sum

 

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3.4. RESULT AND DISCUSSION ........................................................................................................................ 69

 

 

 

 

 

 

 

 

 

3.4.1. Comparison Across the Soil Types ................................................................................................. 83
3.4.2. Comparison Across the Methods ................................................ 85
3.4.3. The Effect of the Propagated Error on the Constitutive Variables ................................................. 91
3.5. SUMMARY AND CONCLUSION .................................................................................................................. 95
APPENDICES .................................................................................................................... 102
A. Pressure-Saturation Measurement Simulation ................................................................................... 102
A.1. Model Conceptualization. ........................................................................................ 103

A2. Model Flowchart .................................................................................................... 104

A.3. Program Listing in Excel Worksheet ..................................... 105

B. Pressure-Saturation Data Obtained Using Saturation Equilibrium Method. ...................................... 138
8.1. Air-Oil Data. .................................... 139

3.2. Air-Water Data. ...................................... . ....................................................................... 142

C. Regression Output. ...... . . . . ...................................... 145
CI. Air-Water System, Based on Data Measured Using Saturation Equilibrium Method. ............................... 146

C2. Air-Water System, Based on Data Measured Using Pressure Equilibrium Method. .. 148

 

 

 

 

 

 

 

 

 

C.3. Air-Oil System, Based on Data Measured Using Saturation Equilibrium Method ..................................... 152
C4. Air-Water System, Based on Data Measured Using Pressure Equilibrium Method. .................................. 145

D. Interfacial Tension Data - -- - - -- ............................................... 157
DJ. Air-Water System - Metea Soil .................................................................................................................. 158
D2. Oil-Water System - Metea Soil .................................................................................................................. 160
D3. Air-Oil System - Metea Soil ........................................................................................... 161
0.4. Air-Water System - Ottawa Sand ........ .. ................................................ 162
D5. Oil-Water System - Ottawa Sand . .............................................................................. 164
0.6. Air-Oil System - Ottawa Sand .............................................................. 165

E. Pressure-Saturation Data Obtained Using Pressure Equilibrium Method .......................................... 166
El. Air-Water Data - Metea Soil. ............................................. 167
13.2. Oil-Water Data - Metea Soil. ...................................................... 177
E.3. Air-Oil Data - Metea Soil. .......................................................................................................................... 185
8.4. Air-Water Data - Ottawa Sand. ................................................................................................................... 188
15.5. Oil-Water Data - Ottawa Sand .................................................................................................................... 198
13.6. Air-Oil Data - Ottawa Sand ................................................................................................ 208

 

viii

Tabl

Iabl

Tabl-

 

 

 

 

 

LIST OF TABLES
Table 2-1: The best estimated air-water and air-oil P-S parameters based

on the pooled data measured using the PE and SE methods .................... 54
Table 3-1: The obtained interfacial tensions, P-S parameters and scaling
factors ....................................................................................................... 70

Table 3-2: Summary of the agreement between prediction outcomes by
methods 2 or 3 to method 1 ...................................................................... 82

ix

 

 

 

ll

 

 

Figure
Figur:

Ilgun

Figun

Figure

Fight

Figure

LIST OF FIGURES

Figure 2-1: Pressure cell detail ................................................................................... 30
Figure 2-2: The apparatus set-up ................................................................................ 31

Figure 2-3: Pressure head at F015 cm and accumulated outﬂow with
respect to time obtained from the PE and SE method
simulations using hp= -50 cmHzO and hp= -37 cmHZO
respectively. ............................................................................................. 39

Figure 2—4: Saturation proﬁles obtained from the saturation equilibrium
and the pressure equilibrium simulations at initial time
(heavy solid lines), during desaturation stage (light solid
lines), at the end of desaturation stage (heavy dash-dotted
lines), during redistribution stage (light dashed line) and at
simulation end time (heavy dashed line). ................................................ 42

Figure 2-5: (a)-(d) Water pressure head changes with respect to time as
recorded by the transducer during redistribution time in a
measurement employing the PE method and (e) the obtained
P-S data from that particular measurement superimposed
with the ﬁtted P—S function by van Genuchten 1981) .............................. 46

Figure 2-6: P—S curves obtained by the PE method (dashed lines)
superimposed with those obtained by the SE method (solid
lines). The conﬁdence limits (shaded area) were based on
regression upon the SE data (diamond marker). ...................................... 49

Figure 2-7: P-S data obtained from measurement using the PE method
(plus marker) and the SE method (diamond marker).The P-S
curves (solid lines) and the conﬁdence limits (dashed lines)
were based on regression upon the PE data (plus marker) ....................... 50

Figure 3-1: Two-phase oil-water and air-oil pressure-saturation curves
obtained by method 1 (solid lines) with their 95% conﬁdence
limits (shaded area) superimposed with those obtained by
method 2 (heavy dashed lines) with their uncertainty limits
(light dashed lines) and the measured data. ............................................. 71

Figure 3-2: Two-phase air-water and oil-water pressure-saturation curves
obtained by method 1 (solid lines) with their 95% conﬁdence
limits (shaded area) superimposed with those obtained by
method 3 (heavy dashed lines) with their uncertainty limits
(light dashed lines) and the measured data. ............................................. 72

 

9.3 2 A... 2.3:...“ w:

 

 

Figure 3-3: Two-phase air-water, oil water and air-oil relative
permeability curves obtained by method 1 (solid lines) with
their uncertainty limits (shaded area) superimposed with
those obtained by method 2 (heavy dashed lines) with their
uncertainty limits (light dashed lines) and the measured data. ................ 73

Figure 3-4: Two-phase air-water, oil-water and air-oil relative
permeability curves obtained by method 1 (solid lines) with
their uncertainty limits (shaded area) superimposed with
those obtained by method 3 (heavy dashed lines) with their
uncertainty limits (light dashed lines). ..................................................... 74

Figure 3-5: Three-phase oil saturation curves obtained by method 1 (solid
lines) with their uncertainty limits (shaded area)
superimposed with those obtained by method 2 (heavy _
dashed lines) with their uncertainty limits (light dashed '
lines) ......................................................................................................... 75

Figure 3-6: Three-phase oil saturation curves obtained by method 1 (solid
lines) with their uncertainty limits (shaded area)
superimposed with those obtained by method 3 (heavy
dashed lines) with their uncertainty limits (light dashed

lines) ......................................................................................................... 76
Figure 3-7: Three-phase oil relative permeability obtained by method 1
(solid lines) and method 2 (dashed lines). ............................................... 77

Figure 3-8: Three-phase oil relative permeability obtained by method 1
(solid lines) with their uncertainty limits (shaded area) and
those obtained by method 2 (heavy dashed lines) with their
uncertainty limits (light dashed lines). ..................................................... 78

Figure 3-9: Three-phase oil relative permeability obtained by method 1
(solid lines) with their uncertainty limits (shaded area) and
those obtained by method 3 (heavy dashed lines) with their
uncertainty limits (light dashed lines). ..................................................... 79

Figure 3-10: Parameters Sr and n estimated by the individual regression
with their 95% conﬁdence limits ............................................................. 87

Figure 3-11: Two-phase pressure-saturation curves obtained by the pooled
regression (heavy dashed lines) with their 95% conﬁdence
limits (light dashed lines) and those obtained by the
individual regression (solid lines) with their 95% conﬁdence
limits (shaded area). ................................................................................. 94

xi

 

LIST OF SYMBOLS

P-S parameter of van Genuchten (1982) P-S function in two phase system pq.
or of air-water P-S curve.
or of oil-water P-S curve.

or of air-oil P-S curve.
scaling factor.

air-oil scaling factor.
air-water scaling factor.

oil-water scaling factor.

uncertainty limits of parameter or.
uncertainty limits of parameter n.
uncertainty limits of saturation.
uncertainty limits of residual saturation.
Brooks and Carey (1964) pore size index.
porosity.

speciﬁc density of the reference phase.
speciﬁc density of phase p.

interfacial tension.

air-water interfacial tension.
oil-water interfacial tension.
air-oil interfacial tension.

oil-water interfacial tension.

accumulated outﬂow volume.

target outﬂow volume in simulation.

contact angle.
covariance matrix [C]=(FT.F)'1

xii

 

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E

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S

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as(h,-) BS(h,-) BS(h,-)]
3Sr 8a an

grain size scale factor in similar media concept.

jacobian matrix [F]: |:

degree of ﬁeedom.
gravitational acceleration
pressure head.

air phase pressure head.

capillary pressure head of air-water system.
capillary pressure head of air-water system.
oil phase pressure head.

capillary pressure head of air-water system.

prescribed pressure head lower boundary condition.
capillary pressure head in two-phase system pq.
water phase pressure head.

summation index, i,j=1, 2, 3, . . .

Leverette J function.

saturated relative permeability.

saturated permeability tensor.

relative permeability.
oil relative permeability in air-oil system.
oil relative permeability in oil-water system.

oil relative permeability in three-phase system.
relative permeability of phase p.

water relative permeability in three-phase system.

water relative permeability in air-water system.

water relative permeability in oil-water system.

P-S parameter of van Genuchten (1982) P-S function in two phase system pq.
P-S parameter of van Genuchten (1982) P-S function in two phase system pq.

capillary pressure.

xiii

 

S58

S 80W

— 30W

St

Saow
W

Saw
W

— 30W

Sw

phase type, p = air, oil, water.
displacement capillary pressure.
wetting-phase pressure.

capillary pressure in two-phase system pq.
nonwetting-phase pressure.

two-phase system p==oil, water. q=air, oil.
wetting phase ﬂux.

macroscopic radius of curvature.

microscopic radius of curvature, radius of curvature interfaces in capillary
tube.

degree of saturation.
degree of saturation as a function of capillary pressure in two phase system pq.

saturation as a function of macroscopic radius of curvature, pore size
distribution function.

water saturation proﬁle at td.
water saturation proﬁle at tc.

water saturation proﬁle at to.

effective saturation S,,E(S-Sr)/(1-Sr ).

oil phase saturation in air-oil system.

oil phase saturation in three-phase system.
degree of saturation of phase p.

residual saturation, residual saturation of wetting phase p in two phase system
P9-
sum of squared error

total liquid saturation, saturation of oil phase plus water phase in three-phase
system.

total liquid effective saturation in three-phase system.
water phase saturation in three phase system.

water phase saturation in air-water system.

water phase effective saturation in three-phase system.

xiv

 

water phase saturation in oil-water system.

time.

time required to drain Vd, drainage time.

student t value at 95% probability and degree of freedom df.
time required to reach equilibrium.

initial time.

direction in Cartesian coordinates.

vertical distance.

XV

 

CHAPTER I

INTRODUCTION

1.1. BACKGROUND

Volatile organic contaminants (V OCs) are extensively present in waste dumps.
Some of the V005 exist in the form of non-aqueous phase liquid (NAPL). When NAPL
is allowed to inﬁltrate the soil, some fraction will partition into and contaminate the water
phase that exist in soil pores. Due to its immiscible nature with water and gas, the
remaining fraction will stay as a separate liquid phase. This fraction will migrate further
leaving a trail of residual NAPL in the soil pore. This residual fraction will become a
long-term source of ground water contamination and further NAPL migration will extend
the contaminated region.

As a special case in immiscible ﬂow problems, the coexistence of gas-NAPL-
water in porous medium is referred to as a three-phase system. The phases involved in
such a system are commonly referred to as air, oil and water phases with the VOC being
the oil phase. According to Gee et a1. (1991) the development of a mathematical model to

accurately assess the spatial and temporal quantiﬁcation of these contaminants is impeded
by the lack of three-phase constitutive information relating relative permeability (kt),
saturation (S) and capillary pressure (P). As pointed out by Abriola (1989), the primary
difficulty is in measuring the capillary pressure of an individual phase in a three-phase

system. Therefore, most analyses of constitutive properties governing three-phase ﬂow

 

2

(Corey, et al., 1956; Stone, 1973; Azis and Settari, 1979) are based on predictive methods
which extend the information obtained from two-phase measurements.

The predictive method conceptualizes the three-phase system as two separate
interdependent two-phase systems; air-total liquid and oil-water. The degree of total
liquid saturation (ie. the saturation of the oil and water altogether) is assumed to be

dictated only by the air-oil capillary pressure and that the degree of water saturation is

is

dictated only by the oil-water capillary pressure. Thus, a complete set of two- and three

phase kr-S-P relationships can be determined based only on the two-phase P-S

information. Parker et al. (1987) proposed a scaling theory which simpliﬁes the
implementation of the predictive method even further. Within the same porous medium,
the scaling theory permits prediction of the PS relationship of any two-phase system
from the P-S relationship of a particular two-phase system provided that the pertaining

scaling factor is known. The scaling theory determines the scaling factors from interfacial

tension data. Therefore, when the P-S data of one two-phase system is available the kr-S-

P information involving any ﬂuids in any combinations can be determined. The
additional data required are only the interfacial tension data.

The scaling theory was incorporated in a versatile numerical three-phase ﬂow
model (Kaluarachi et al., 1990), which was further improved and has been shown valid
for both desaturation and imbibition scanning paths (Parker and Lenhard, 1987b) and to
accept multi component organic chemical transport (Parker and Kaluarachchi, 1990).

This model is one among the few formulated to predict DNAPL movement (Huling and

 

3

Weaver, 1991). Incorporation of scaling theory into the model eliminates the necessity to
provide air-oil and oil-water P-S information. It allows the use of air-water PS and
interfacial tension data only.

NAPL contamination assessment in the unsaturated zone most likely involves
unconsolidated soils. Air-water P-S measurement on an unconsolidated soil sample is a
common practice among soil scientist and the measurement protocol for this case is well
established. Thus, air-water P-S data are commonly available. Interfacial tension
measurement can also be considered standard measurements with an established protocol
as described by Adamson (1976). Considering that the data requirements are less, scaling
theory greatly improves the applicability of the multi-phase and multi-component model
by Kaluarachi et al. (1990).

Beside the advantages stated above, however, the use of scaling theory in the
predictive method may contribute to additional loss in accuracy of the obtained kr-S
relationship as Lenhard etal. (1987) has suggested. Scaling theory regards the soil matrix

as a rigid porous medium. Accordingly, the pore size distribution is also assumed

constant with respect to time. This assumption is widely adopted in the arcs of ground

water engineering. However, as noted in Lenhard et al. (1987), the k,-S relationship

prediction via scaling theory might be sensitive to any disturbance in the pore size

distribution. Small changes in pore size distributions, such as that due to swelling, which

is not clearly evidenced in the P-S plot might aﬁ'ect the prediction of k,-S curves

signiﬁcantly. On the other hand, the pore size distribution is inherent in P-S relationship

4

and this relationship is represented only by the P-S parameters. Therefore, any error or

bias in these parameters might also affect the prediction of kr-S curves signiﬁcantly such

as the effect of pore size distribution changes. Note that, kr-S-P relationship prediction via

scaling theory is based on P-S parameters measured in one two-phase system only. These

parameters are referred to as base parameters. The above discussion suggests the

importance of a study to investigate the effect of error or bias embodied in the base
parameters to the prediction of the k,-S-P curves via the scaling theory.

The magnitude of error and the potential of obtaining bias parameters can be
deduced from the precision of those parameters. Accuracy and precision in determining
the P-S parameters might be affected by soil texture and the type of the immiscible ﬂuid
involved. A study by McCuen et al. (1981), based on 1,085 samples measured by Rawls
et al. (1976) and Holtan et al. (1968), observed distinct P-S parameter variance between
soil texture classes. As observed in the measurement result reported by Lenhard and
Parker (1987), the P-S parameters derived from air-water data possessed higher variation
than those determined from either air-oil or oil-water data . If the accuracy and precision

of the P-S parameters differs from one two-phase or from one porous medium to the

other, so does the performance of the prediction of the k,-S-P relationship. In turn, the

scaling theory might perform differently when it is implemented on different soil texture

using base parameters from different two-phase system.

5

In providing the three-phase kr-S-P relationship information the predictive method

relies on the availability of two-phase P-S information. When involving unconsolidated
soil samples, which is typical for subsurface soil samples, P-S measurement of the
air-water system is more common than the air-oil or oil-water system. Therefore,
air-water P-S data in general is widely available. When information of air-oil or oil-water
P-S relationship are required, in most cases, measurements should be conducted. On the
other hand, established P—S measurement, such as that by Klute (1986), are primarily
intended to measure air-water system. The characteristics of an air-water measurement
protocol are: suitable only for a low wetting phase volatility, limited to gaseous type of
non-wetting phase, and takes a considerable amount of time. In parallel with current
growing interest in the multiphase ﬂow problem, it is desirable to have an efficient P-S
measurement technique that measures not only air-water systems but also air-oil and
oil-water systems.

Conventionally, P—S data are measured by allowing saturation to equilibrate under
a series of ﬁxed pressures imposed at the boundaries. While this technique is considered
to be standard, a considerable amount of time is required to measure the entire P-S
relationship. Typically, time required to obtain sufﬁcient data to adequately represent the
desaturation scanning path is approximately two weeks (Su and Brooks, 1980; Bear,
1982). Such a disadvantage signiﬁcantly constrains the gathering a large amount of data,
sample replication, and to measure data near the region of residual saturation. Yet, it is

desirable to include these measurements. Replication and a large number of data insure a

 

6

representative sample mean as well as knowledge about the variance. Inclusion of data
from the region near residual saturation reduces the risk of bias.

The time required to measure P-S data using the conventional method is primarily
dictated by the time required for the system to reach hydrostatic equilibrium. Evidently,
any measurement readings in other than the hydrostatic equilibrium state would not
represent the primary drainage or imbibition P—S relationships. (Topp et al., 1967; Smiles
et al., 1971; Carey and Brooks, 1975; Parker et al., 1985). A P-S measurement method
devised by A. T. Corey, as reported by White et al. (1970) is capable of reducing the
measurement time while obtaining data that are representative of primary drainage. This
technique reverses the conventional method by requiring capillary pressure to equilibrate
under a series of ﬁxed externally imposed saturation. Henceforth in this dissertation
(based on the phenomenon to deduce the equilibrium state of the sample), the
conventional method is referred to as the saturation equilibrium method. The method
devised by Corey is referred to as the pressure equilibrium method. The pressure
equilibrium method has been employed by White et al. (1970) and White et al. (1972)
and was improved upon by Su and Brooks (1980). Typical total time required to measure
9 data points in air-water desaturation and imbibition scanning paths was less than 24
hours (Su and Brooks, 1980).

While the pressure equilibrium technique has been shown capable of reducing the
measurement time signiﬁcantly, the obtained data have never been formally compared to
that obtained using the conventional method. As such, the time reduction achieved may

only be the result of a premature reading which might be biased relative to the

7
conventional measurement. If the converse was true and experimental time was indeed

shortened, this technique offers an efficient method which might be ﬁrrther improved to

measure any type of immiscible ﬂuid.

1.2. GENERAL OBJEcrrVEs, APPROACHESAND PRESENTATION
ORGANIZATION

This study focuses on the effects of incorporating scaling theory in the predictive

method to construct the three-phase kr-S-P relationship. Without using scaling theory, the

P-S parameters required by the predictive method should be determined from P-S
desaturation data of air-water, oil-water and air-oil systems. With scaling theory,
however, only one of those systems is required to be measured. This study investigates
the alternative of using either the air-water or air-oil systems as the base system. The
air-water system is not directly involved in a three phase system. However, air-water data
are commonly available. Conversely, the air-oil system involves in a three-phase system
but P-S data of this system are not as widely available as that of air-water. The
performances of these alternatives are investigated in two soil textures. The textures
chosen are sand with uniform grain size distribution and loamy sand. Lenhard and Parker
(1987) proposed the scaling theory as a way of improving the efﬁciency of the predictive
method. Accordingly, the discussion throughout this dissertation is in the context of
testing the hypothesis that the predictive methods with and without the scaling theory are
not different regardless of the usage of the two-phase system to obtain the base

parameters and the involvement of soil texture.

 

8

Additionally, this study veriﬁes the pressure equilibrium measurement technique
against the conventional measurement method as part of an effort of developing an
efﬁcient P-S measurement technique. The Su and Brooks (1980) apparatus is modiﬁed
and improved to make it suitable for air-water, oil-water and air-oil P-S desaturation
measurement. The concept of the modiﬁcation is veriﬁed theoretically. An unsaturated
mathematical model based on Richard’s equation is developed to simulate P-S
measurement employing the pressure equilibrium method as well as the saturation
equilibrium method. The modiﬁed apparatus is used to measure P-S data of air-water and
air-oil systems. The pressure equilibrium is proposed to provide an efficient two-phase
P-S measurement technique substituting the conventional saturation equilibrium method.
As such, discussion in this dissertation is focused to observe whether the pressure
equilibrium method is capable of reducing the time required to measure two-phase P-S
data while conserving the characteristic of the primary desaturation.

The two major issues addressed pertain to the implementation of the scaling
theory and the veriﬁcation of the pressure equilibrium measurement method. These two
issues are discussed in Chapter II and Chapter III. Each of these chapters is presented as a
self-contained section. Each one contains a concise discussion of the background,

description of the material and method used and subsequently discusses the results and
ﬁndings speciﬁc to the particular issue. Experimental protocols, measurement result, data
reduction and statistical calculations are presented in the appendices.

Background of the issue is presented in three separate sections. First, the

Deﬁnition and General Overview section in this chapter clariﬁes the terminology used

9
and summarizes the major existing concepts pertaining to the pressure-saturation scaling
theory. It is intended to draw a frame work from a broad point of view. Additional
background, more speciﬁc to the separate issues is presented concisely in the introductory

sections of Chapters 11 and 111. These sections are intended to put the addressed issues in

perspective.

1.3. DEFHVITIONSAND GENERAL OVER VIEW.

As mentioned above, this dissertation is concerned with the scaling method in the
context of its implementation in the predictive method. The predictive method is a
concept to provide an alternative means for solving multiphase ﬂow equations. Prior to
discussing the existing methods to scale the P-S relationship, the subsequent sections
describes some terminology pertaining to multiphase ﬂow used in this dissertation.
Following that, the involvement of the predictive method in solving the multiphase ﬂow
is presented. The last section reviews the existing methods to scale the P-S relationship
and highlights the porous medium hydraulic properties that are postulated by some

authors as being the factors in the scaling method. The notations used in the presentation,

generally follow the symbols used in Parker (1989).

1.3.1. Multiphase System
When two or more ﬂuids coexist in a soil pore and one or more of those ﬂuids are
in motion, the types of ﬂow possible are either miscible or immiscible displacements

(Bear, 1972). Miscible displacement refers to the case where the ﬂuids are completely

 

10
soluble in each other. A distinct ﬂuid-ﬂuid interface which separates those ﬂuids does not
exist. In the immiscible displacement, each ﬂuid for practical purpose exists separately.
As such, a distinct interface between ﬂuids exists and the interfacial tension between the
two ﬂuids is not zero.

Fluid motion accompanying immiscible displacement is commonly referred to as
multiphase ﬂow. In this dissertation the term phase is used speciﬁcally to refer to the
ﬂuids involved and not the soil grains. A multiphase system refers to the coexistence of
two or more phases in soil pore in conjunction with the multiphase ﬂow concept. A
multiphase system in general may involve two or more ﬂuid types. However, a
conceptualization of multiphase system which considers only three types of ﬂuid is
commonly regarded as sufficient and representative as the most general form of a
multiphase system. This dissertation uses the term three-phase system in this context. In
conjunction with an interest in subsurface contamination assessment, the three ﬂuids
involved in a three-phase system are oﬁen classiﬁed as water, NAPL and gaseous phases.
The NAPL and gaseous phase are commonly referred to as oil and air phases
respectively. Henceforth, the terms air, oil and water are used to refer to the phases
involved in the three-phase system.

A two-phase system is a multiphase system involving only two phases. The
phases in a two-phase system are referred to as wetting-and nonwetting. The phase that
tends to have direct contact with the solid is referred to as the wetting phase. The two-
phase system consisting of air and water is a special case. This system describes

condition in the unsaturated zone prior to contamination. The water and the air phases are

11

the wetting— and nonwetting-phases, respectively. Unsaturated ﬂow is a particular case of

two-phase ﬂow. In this case, the interest is focused on the water phase motion only and

the air phase is assumed stagnant at atmospheric pressure.

1.3.2. Multiphase Flow Equation and the Predictive Method.

The equation governing the multiphase ﬂow is derived ﬁom the generalization of
Darcy’s equation combined with the continuity equation. (Bear, 1972; Corey, 1982;
Parker, 1989). According to Parker (1989), the governing equation of the multiphase

ﬂow in a non-deformable porous medium, under the assumption of no sources or sinks of

the phases involved, is given by:

¢a(pp ___p_S=) 3 k kn[3_:p a_z_]
3"!

— x,'Pp tr 11 ""pp ax

The subscript p is reserved to indicate the ﬂuid phase (p=a, o, w) while i and j indicates
directions (i, j=1 ,2,3) with repeating values indicating summation in tensor notation, x is
the Cartesian coordinate, z is the vertical elevation, 4) denotes the porosity of the medium,
I is time, pp is the phase density, km is the phase relative permeability, k,- is the intrinsic
permeability tensor, Sp is the degree of saturation by the phase which is the fraction of the
soil pore occupied by phase p, and hp is the pressure head of phase p.

Note that h, S and k1. are interrelated. When capillary pressure h increases the
saturation is decreased. Saturation decrement reduce the pore volume that participate in

providing the channel for the ﬂuid in question to ﬂow. The relationship between kr-S-h is

12

formulated in the constitutive equations. The constitutive equation goveming multiphase
ﬂow consists of two functional relationships. The ﬁrst relationship is a function relating
the saturation of a particular phase and the pressure heads of the phases involved. The
second is the relationship between the relative permeability and the saturation. The ﬁrst
function is referred to as pressure-saturation (P-S) function. In a three-phase system, this

function is formulated as:

Sp=f(ha’ ho, kW) [1.1]
The P-S function in a two-phase system is a special case that involves only the pressures
of the wetting and nonwetting phases. Interaction between these pressures can be

represented only by the capillary pressure which is deﬁned as:

PM 5 Pp — Pq
Here PM is the capillary pressure, Pp is the pressure of the nonwetting phase p and Pq is
the pressure of the wetting phase q. The capillary pressure Pm is commonly expressed in

terms of pressure head relative to some reference phase as:
m=mem4
where hm is the capillary pressure of the pertaining immiscible phases pq in term of
pressure head, g is the gravitational force, and p“ is the density of the reference phase.
The two-phase P-S ﬁinction will then take a simple form as:

s= 11h”). [1-2]

The procedure to solve the ﬂow equation commonly involves a step to reduce the

nonlinearity of that function by applying the chain rule to the term of the derivative of

13

saturation with respect to time. The form of the chain rule for the case of three-phase ﬂow
is written as:

as, as, ah, as, ah, as, ahw
= 1-3
at ah, at + ah, at + ahw 6t [ 1

This means that the constitutive equation that describe the relation of oil saturation and
the capillary pressure heads should be formulated in the form of Eqn 1-1. Construction of
such a relationship requires direct measurement of the saturation and the pressure head of
the air, oil and water phases simultaneously. This requires three phase measurements
which are not commonly made. Furthermore an established protocol and technique to
conduct three-phase P-S measurement is not yet available. Lenhard and Parker (1988)
are the only investigators to report three-phase measurements for P-S relationships.

The common method for solving the three-phase ﬂow equation is by employing
the predictive method proposed by Leverett (1941). The predictive method assumes that
the soil matrix is a hydrophilic porous medium. Under this assumption, the water phase
has a direct contact with the soil grain while the oil phase always resides between the
water and the air phases. Provided that the oil volume is high enough to cover the water
surface, the air phase would have no direct contact with the water phase. The predictive
method further assumes that the total liquid saturation (ie. oil plus water) is dictated only
by the air-oil capillary pressure and that the water saturation is dictated only by the
oil-water capillary pressure.

Employing the above point of view, a three-phase system consisting of air, oil and

water can be conceptualized as two interdependent two-phase systems. The ﬁrst system is

 

14
a two-phase system consisting of a total liquid phase and an air phase. Since the
saturation of the total liquid is a function of the air-oil capillary pressure only, the total
liquid-air system is assumed to be identical to an air-oil system. Secondly, within the total
liquid phase, the oil-water two.phase system exists. Employing this approach, the above
chain rule would only involve the derivative of total saturation with respect to air-oil
capillary pressure and derivatives of water saturation with respect to oil-water capillary
pressure. Therefore, it only requires knowledge of the air-oil and oil-water P-S

relationship. These relationships are easier to measure compared to a true three-phase P-S

relationship.

1.3.3. Methods to Scale the Pressum-Saturation Relationship.
”“2 [E l' Elf. -S 'Sl'

P-S scaling theory was developed to relate a P-S curve across porous media
hydraulic system. This knowledge allows transformation of the P-S relationship from
certain system to another system. Prerequisite to this knowledge is an understanding of
any determinants that dictate the P-S relationship. According to Bear (1972), because of
the dependency of capillary pressure P on interfacial tension 0 and radius of curvature R,

on a microscopic level, macroscopically it depends on (a) the geometry of the void space,
(b) the nature of the soil-ﬂuid interaction and (c) the degree of saturation S. On a
macroscopic level, the porous medium property associated with the characteristic of the

void space geometry is the pore size distribution. The factor of soil-ﬂuid interaction

includes contact angle (.0 as suggested by Morrow (1976) and Demond and Roberts

15

(1990) among others and wettability as suggested by Scheideger (1969) and Adamson
(1967). The study by Anderson (1987), however, concluded that the PS relationship was
insensitive to the wettability at large saturation ranges as it is overwhelmed by the effect
of surface roughness. This dissertation assumes that the effect of wettability is
insigniﬁcant. The third factor which is the dependency of capillary pressure to the
saturation, is formulated in the form of P—S relationship curve. Therefore, it is obvious
that inherent in the P-S relationship there exist interactions between P, R,, 0', co and S.
The following paragraphs discuss further the conceptualization of these interaction.
Capillary phenomena in a soil matrix are commonly explained based on Laplace’s

equation. Under this concept, the capillary pressure Pp, is related to the radius of

curvature Rc of the interface by:
2 a

Note that, a P-S relationship describes the relationship between the capillary
pressure and saturation at a macroscopic level. Hence, a given capillary pressure does not
necessarily refer to a particular radius of curvature in a particular soil pore. On a
macroscopic level, R, is rather the radius of a conceptual capillary interface in the soil-
ﬂuid system that exits as the response of the system when subjected to that capillary
pressure. To distinguish between the microscopic and the macroscopic radius of curvature
let the macroscopically measure be denoted as R. Carey (1982) deﬁned R as the hydraulic
radius equivalent to that commonly used in hydraulic engineering. Bear (1982) put this R

in the context of “statistical average taken over the void space in the vicinity of a

 

16

considered point in the porous medium”. The most important characteristic of R is that it
decreases as the capillary pressure Pp, increase and when R decreases S decreases
proportionally. The conceptual relation between S and R is commonly associated with
pore size distribution and is taken to be a porous medium dependent parameter. In this
dissertation the term of pore size distribution function is used speciﬁcally to refer a
function that relates S and R in the form of R(S).

Based on the conception above, a P-S relationship in the form of saturation as a
function of capillary pressure 5(qu) can be factored into two relationships. The ﬁrst
relationship relates the capillary pressure and the radius of curvature Pp,(R). This
relationship is formulated following the concept behind Equation 1-4. The second relates
the macroscopic radius of curvature and a function of saturation which is conceptualized
as the pore size distribution function R(S). Therefore, the function of S(Pp,) can be

viewed as the result of substituting the R(S) into Pp,(R).

Following this conceptualization, an S‘( P3,) curve of soil-ﬂuids system A can be
transformed to become 53(P;q) of soil-ﬂuids system B by scaling their capillary
pressures such that at any given S there exist a relation of P3, (R) = [3(5) Pg, (R). The
factor B is referred to as scaling factor and hence is deﬁned as:

13(8); Pym/Pia). [1-51

The general formulation of the scaling factor can be obtained by substituting Equation 1-

4 into Equation 1-5 and is written as:

17

a” R‘(S)
s = — 1-6
[OM . 1

Within a soil-ﬂuid system the interfacial tension is commonly regarded as a

 

constant. The macroscopic radius of curvature R, however, varies as the saturation
changes. When the saturation in the system A and B is changed with the same amount,
the R in systems A and B is not necessarily changed with the same amount. Therefore, the
ratio RA/RB varies when the saturation is varied. Equation 1-6 accommodates this
characteristic by expressing the macroscopic radius of curvature R as the pore size
distribution function R(S). Therefore, to obtain a scaling factor that transfonn P-S curves
across two different two-phase systems and porous media requires knowledge of the
interfacial tensions and pore size distribution functions.

The scaling factor as formulated in Equation 1-6 might obscure its practical value.
To date, most common technique to deduce the pore size distribution function requires
P-S data When P—S data are measured and available for both systems A and B, scaling
procedures are not required. Some investigators have proposed relating the pore size
distribution function to the grain size distribution. For example, Smith (1933) and White
et al. (19703) simplify the shape of the soil grain into sphere shapes to formulate the void
structure. Even after such simpliﬁcation, the results were still too complex for practical
use and yet too speciﬁc to serve any general purpose. Apparently, procedures for scaling
the P—S curve are only feasible within identical soils or at least within similar porous

media as deﬁned in the similar media concept by Miller and Miller (1956).

18
WW
Leverette (1941) proposed a function, referred to as .1 function, to represent a

general form of the P-S relations. This function is given as:

Mara

Here k is the saturated permeability and (p is the porosity. In his report, Laverette (1941)
showed that Equation 1-7 coalesced the P-S curves measured in several unconsolidated
sands. Based on this notion, the J function is commonly regarded as function that can be
used to scale P-S curves not only across the two-phase system but also across porous
media.

To scale P-S relationship, Equation 1-6 utilizes a factor which transforms P-S
relationship of a particular soil-ﬂuids system to another. The J function follows different
approach. This method transforms any P-S curve into its stande form, which is the J
function as written in Equation 1-7. In order to compare the scaling concept behind
Equation 1—6 and Equation 1-7, the scaling factor is reformulated based on Equation 1-7.
The factor which scales the capillary pressure in system A into system B at a given J(S"')
is given by:

"7
B = 9;. 8A [1-8]
a k3
A.

' According to Bear (1972) and Corey (1982), based on the concept of relative

 

permeability proposed by Burdine (1956), the term of (k,/<l))”2 is equivalent to R at

 

 

ll

19

saturation in which the unsaturated permeability is k,. When kr is substituted by the
saturated permeability k, the term (k/¢)“2 can be interpreted as the macroscopic radius of
curvature when the effective porosity are fully ﬁlled with the wetting phase. If this

particular radius of curvature is denoted as Rm, , Equation 1-8 becomes:

>

OB R
_____._ﬁ 1-9
B a, R r 1

B

ax

Be:

When comparing Equation 1-9 with Equation 1-6, it shows that the J function scales the
P-S relationship across porous media using a constant (the ratio of Rm“) rather than using
the pore size distribution function. It is expected that this particular scaling method will
not always hold. In a later study using air-water system in various consolidated and
unconsolidated porous media (Rose and Bruce, 1949), shows that plots of the J function
are unique to the porous medium type. Based on this, further studies (Rapoport 1955;
Richardson, 1961) that advocated the use of this function to scale the P-S curves,
restricted the use of their method for identical porous medium. Under this restriction the
ratio of Rm, becomes unity. In turn P-S scaling by J function accommodates the scaling

across two-phase systems only and excludes the feature of across porous media.

liii S' 'l M l' E 1 Mil lll'll 1222!

Miller and Miller (1956) proposed the similar media concept as a method to scale
the unit of a physical model to simulate unsaturated ﬂow in porous media. Here two
porous media are considered as similar when the grain size axis of their grain size

frequency distribution curves can be scaled to each other by a constant factor d.

 

20

According to this method, all P-S curves within similar media will coalesce into a

standard curve when their capillary pressures are scaled by:

P=[&9—]d {1.10}

where P is the scaled capillary pressure. Using superscript A and B to denote the oil-

ﬂuids systems, the scaling factor equation derived based on Equation 1-10 will take form

of:

oB dA

“-077

[1-11]

This formulation is equivalent to that formulated in Equation 1—9 where the scaling across
porous media is accommodated by including the term of the ratio of d. It should be noted,
however, to satisfy the similar media condition the two porous media should have similar
grain size frequency distribution. Inherently, this condition requires that the two porous

media have similar shape pore size distribution curve. When this condition is applied in
Equation 1-6, the term of the ratio of the pore size distribution functions will be constant.
As the result the scaling factor B in Equation 1-6 will take form of constant rather than
function. Thus, the scaling factor as formulated in Equation 1-6 is in agreement with that
in Equation 1-11.

Study by Klute and Wilkinson (1958) conﬁrmed that when the criterion of similar
media was satisﬁed P-S curves of several different-but-similar sands were coalesced. The
study by Elrick et al. (1959) using soil containing not only sand fractions but also silt and

clay, however, concluded that when the theory was checked against condition far from

21

ideal a deviation was observed. Moreover, this study demonstrated the sensitivity of the

scaling procedure to any pore size distribution variation.

.v

Hi, 41.! I a

A special case of interest in scaling the PS curve is when this curve is expressed

following the equation proposed by Brooks and Carey (1964). Here the P-S relationship

is formulated as:
P A
s, = [—4-] [1-12]
PP?

where S, is the effective saturation which is deﬁned as (S-S,.)/(l-Sr ), Sr is the residual

saturation, Pd is the displacement pressure and A is the pore size index. All the required

parameters to scale the P-S curve are already inherent in this formula since all parameters

involved have a clear physical interpretation. When S is expressed in term of S, the

differences between the two P-S curves due to their differences in Sr are factored out.

Thus P-S curves from any two-phase system will have common range of S, from 1 at full

saturation to zero at residual saturation.

The displacement pressure Pd is the capillary pressure associated with the largest

pore in the soil-ﬂuid system in question. When the capillary pressure in a P-S plot is

factored by Pd, a standard capillary pressure equivalent toPused in Miller and Miller

(1956) is obtained. The parameter It. represents the pore size distribution characteristic

 

22
and was postulated to be a porous medium dependent parameter. In turn, any two P-S
curves that have identical 2. will satisfy the requirement for similar media. Therefore, if
those two curves are expressed in term of effective saturation and reduced capillary

pressure they will coalesce into a single curve. Based on this particular characteristic, the

parameters Pd and A are often used as parameters upon which the characteristic of the

scaling method is observed such as in studies by Demand and Roberts (1990), Lenhard
and Parker (1987) among others.

Even though P-S formulation given by Eqn 1-12 provides all properties required
for scaling procedure, the use of this function to conduct scaling across porous media is

not practical. When a scaling equation comparable to Equation 1-6 is derived based on
Equation 1-12, the resulting equation will be a function of A. This means that the

knowledge of pore size indexes of the porous media involved is required. As mention

previously, to date, these indexes can only be obtained by means of P-S measurement.

manmmammsmzmm

As part of their effort to develop an efficient method to construct a three-phase
constitutive relationship, Lenhard and Parker (1987) proposed their scaling theory. This
method was intended to scale the PS curves across two-phase systems within a single
porous medium. Therefore, their scaling factor took the form of Equation 1-6 in which
the term of RA(S)/RB(S) is set to unity since the soil in systems A and B are identical.

Further detail description of their theory is presented as part of Chapter III.

 

23

Waugh.

In its original form, the contact angle a) is involved in the Laplace equation.

When (D is included, Equation 1-1 takes the form:

a” case)” )[R‘(S)] [1-13]

[3(5) = [0* -cosa)A R"(S)
Among investigators, the inclusion of contact angle a.) is still a point of controversy.
Several investigators who worked with artiﬁcial unconsolidated porous medium (Bethel
and Calhoun, 1953; Morrow, 1976; Demand, 1988; and Demand and Roberts, 1990)
observed the effect of a) on the P-S relationship. Subsequent work (Demand and Roberts,

1991) on a sand-organic liquid-water system showed the discrepancy between the
observed and scaled displacement pressure when (u was excluded.

Exclusion of a) (Leverette, 1941; Rapoport, 1955; Parker, 1987) was based on the
argument that conceptualization of soil pore geometry as a tube was not appropriate.

Hence, the radius of tube should be expressed in term of radius of curvature.

Arithmetically, when the radius of tube is expressed in term of radius of curvature the
contact angle a.) will be canceled out. By not including contact angle to, Dumore and
Schols (1974) were able to successfully scale their data of heptane-water and toluane-
water in consolidated sample. Anderson (1987) observed that both in strongly water-wet
and oil-wet consolidated rock sample, the capillary pressure is insensitive to a) at large

range values of (0, especially in the drainage path.

24

It is interesting that most of the investigations that reported an observable effect of
a), used an artiﬁcial unconsolidated porous medium or clean sand. On the connary, those
who used natural rock core or unconsolidated soil concluded that due to the existence of

surface roughness, the apparent effect of contact angle is almost non-existent. In this

dissertation the effect of contact angle co is assumed negligible.

[SIZE 1 CSI' Mil $11.1
Several studies indicate the dependency of the parameters Pd and A on soil

texture. This dependency is based on the assumption that in an air-water system, the pore
size distribution characteristic is strongly affected by the grain size frequency
distribution. Several studies even advocated deduction of the P-S curve based on the
parameters associated with the grain size distribution (McCueen et al., 1981; Campbell,
1985; Mishra et al., 1989 among others).These approaches were based on regressing an
empirical function onto grain size distribution data. Brakensiek et al. (1981) used the

Brooks-Corey (1964) equation as their regression function. In this study the parameters
Pd and A were classiﬁed into the twelve Soil Conservation Service textural classes. A
study by Cosby et al., (1984) conﬁrmed that, statistically, the magnitudes of Pd and A

were strongly dictated by the soil texture. Similarly, the study by McCueen et al., (1981)

found that these parameters are unique for each soil texture class.

CHAPTER II

THE A CCEPTABILITY OF THE PRESSURE EQUILIBRIUM METHOD
To MEASURE DRAINA GE PA TH 01“ San PRESSURE-SA TURA TION

2.1. BACKGROUND

The pressure-saturation (P-S) relationship is one of the constitutive relationships
governing multiphase ﬂow through unsaturated porous media. It expresses the functional
relationship of the equilibrium wetting phase saturation at a given capillary pressure.
Petroleum engineers employ the P-S relationship (Collins, 1961), while soil scientists
present the same information in a water retention or moisture characteristic curve (Child,
1940). The soil scientist’s term, however, is commonly restricted to air-water systems,
while the more general P-S relationship is employed with a variety of immiscible ﬂuid
systems. In parallel with current growing interest in a multiphase ﬂow problem, it is
desirable to have an efﬁcient P-S measurement method that measures not only air-water
systems but also air-oil and oil-water systems.

Conventionally, the P-S data are measured by allowing saturation to equilibrate
under a series of ﬁxed pressure gradients imposed at a boundary. While this technique is
considered to be a standard, it is a very time,consuming method. The typical time period
required to obtain sufﬁcient data to adequately represent the desaturation scanning path is
approximately two weeks (Su and Brooks, 1980). Such a disadvantage places signiﬁcant
limitations in collecting larger numbers of data, replicating the sample, and measuring the

data near the residual saturation region. Replication and the large numbers of data are

25

 

26
desirable to obtain representative mean and to increase precision reliability. An inclusion
of data from the region near residual saturation eliminates the risk of bias.

The length of experimental time in the conventional P-S measurement is primarily
dictated by the length of time required for the system to reach equilibrium. Up to this
time, studies that investigated the feasibility of conducting measurements during the
transient state have had qualiﬁed success. At a given saturation, capillary pressure
measured under dynamic ﬂow tended to be higher than those in a static condition (Topp
et al., 1967; Smiles et al., 1971; Carey and Brooks, 1975; Parker et al., 1985).
Apparently, in order to obtain P-S data that share similar characteristic to the
conventional method, any alternative method that measures the P-S relationship under an

equilibrium state is more sound.

An alternative method that measures the P-S relationship under the equilibrium
state was devised by A. T. Carey (White et al., 1970). This method had been used in White
et al. (1970) and White et al. (1972) and was improved by Su and Brooks (1980). In
contrast to the conventional method, suction was applied in two stages. In the ﬁrst stage
suction was set to drive a reasonably quick desaturation. While in the second stage, after
a volume of desaturation had been obtained, the suction was adjusted to balance the
equilibrium wetting phase pressure. Implementing this method to measure air-water data

using 50 different soils covering a wide range of properties, Su and Brooks (1980),
observed that typical experimental time to complete both desaturation and imbibition

scanning paths was less than 24 hours. The transient time for the pressure to reach

 

ll

27

equilibrium after the desaturation step took only a few minutes at high saturation and a
few hours at low saturation region.

While this technique signiﬁcantly shortens the experimental time, the obtained
data have never been formally compared to data using the conventional method. As such,
the time reduction achieved may only be the result of a premature reading which could
introduce bias relative to the conventional measurement. If the converse were true and
experimental time was indeed shortened, this technique offers an efﬁcient method which
might be further improved to measure any type of immiscible ﬂuids. Note that in the Su
and Brooks (1980) apparatus, a wetting phase evaporation was allowed to accelerate the
desaturation rate. If the wetting phase is a volatile liquid, the desaturation might be
dominated by evaporation. This condition may introduce a problem in adjusting the
second stage suction. Furthermore, this approach prohibits the use of a liquid nonwetting
phase. However, applying a high positive pressure on the nonwetting phase to desaturate
the sample may resolve those problems. This modiﬁcation, however, is only justiﬁed
providing that it will not affect the characteristics of the method for reaching equilibrium
quickly.

According to Su and Brooks (1980), the equilibrium state was quickly reached
due to the adjustment made in applying the second stage suction. Unfortunately, no
further explanation was given. In their measurement technique this adjustment was
conducted in the following manner. After a ﬁnite volume had been drained from the
sample, the outﬂow was directed toward a vertical capillary glass tube containing the

meniscus of the air-wetting phase interface. At this instance, the air pressure on that

 

28

interface was adjusted to maintain the meniscus level at a prescribed datum until this
meniscus became stationary. Leveling the meniscus during the transient state will permit
determination of the wetting phase pressure. In our opinion leveling the meniscus
changes the condition at the external face of the plate from a constant head to a zero ﬂux
condition. This suggests that the important behavior underlying the Su and Brooks
(1980) method is that the transient state within the soil sample which occurs under a no-
ﬂow boundary condition will be terminated more quickly than it would under a constant
head boundary condition as in the conventional method.

Based on this premise, applying a positive pressure to desaturate the sample
should give no effect to the basic principle of the method. Furthermore, replacing the
capillary tube with a zero displacement pressure transducer seems to be more effective in
providing a zero-ﬂux boundary condition while monitoring the wetting phase pressure.

F urthermore, the use of a transducer eliminates the laborious second stage suction
adjustment. If the transducer used is capable of measuring the pressure difference
between the wetting and nonwetting pressure, a more efficient measurement protocol can
be obtained.

In order to distinguish between the Carey method and the conventional method
based on the phenomenon used to indicate the equilibrium state of the system, the Carey
method will be referred to as the pressure equilibrium (PE) method and the conventional
one as the saturation equilibrium (SE) method. The current paper studies the implementa-

tion of the PE method by developing a P-S measurement apparatus based on the concept

referred to in the previous paragraph. The apparatus developed was tested to measure

29

air-water, air-oil and oil-water desatmation P-S relationships in a natural loamy sand soil.
To justify the acceptability of the PE method, the air-water and air-oil data obtained were
compared to those obtained by the SE method. In addition, a 1-D unsaturated ﬂow

mathematical model was used to verify the method theoretically.

2.2. MEASUREMENT METHoD AND MATERIAL

The PE method measures the P-S desaturation data by imposing a ﬁnite
desaturation on a presaturated sample and subsequently allowing the ﬂuids in the sample
to equilibrate under a no-ﬂow boundary conditions prior to recording the equilibrium
data. The equilibrium state in this case is indicated by a stationary wetting-nonwetting
pressure difference. With the SE method, desaturation is allowed to occur under a ﬁxed
wetting-nonwetting pressure difference imposed at the boundaries until equilibrium is
reached as indicated by an outﬂow dirninishment. In order to compare the performance of
these two methods, a P-S measurement device capable of performing either PE or SE
methods in measuring air-water, oil-water or air-oil desaturation data was developed.

This device was developed utilizing a standard pressure cell by Soil Moisture
(Model No: 1400) as shown in detail in Figure 2-1. The connection between the pressure
cell and the other parts of the device is shown in Figure 2-2. Three pressurized ﬂuid
sources are available: air, oil and water. The air phase source is obtained from the
building line through a cascade of coarse and ﬁne regulators. The oil and water phase
pressure chambers provide the source of the oil and water phases respectively. The

pressures of these liquids are governed by the air pressure in the chamber. This pressure

30

 

 

 

: 1. Top assembly
4; 2. Inﬂow port

3. Filter paper
4. Wing nut

 

5. Wm O-ring

    

———> 6. Soil sample

-——> 7. Brass retaining
cylinder

———> 8.Filterpaper

 

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9. Viton O-ring
12. l-baggiggrlme % 10. Bottom assembly
plate ; 11. Outﬂow port

 

 

 

 

Figure 2-1: Pressure cell detail

31

 

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32

can be altered utilizing the building pressurized air. The zero pressure of these liquids is
deﬁned as the pressure which level the air-liquids interface in burettes B1 and BZ (Figure
2-2) at the center of the cell. The applied building air pressure at any instance is
monitored via a mercury manometer.

The connections between the ﬂuid sources and the cell are designed such that the
air and water phases can be used as the nonwetting and the wetting phases respectively,
while the oil phase may function as either one. The nonwetting phase line is connected to
the top of the cell. The positive pressure applied on the nonwetting phase drains the
wetting phase from the bottom of the cell. The outﬂow is directed into a vial collector in
which the pressure is atmospheric. The outﬂow line to the vial is controlled by a valve,
referred to as the collector valve. The collector valve is opened only when desaturation is
permitted. A differential pressure transducer is installed to measure the pressure gradient
between the wetting and the nonwetting phase pressures.

Typical procedures in conducting measurement are: packing the sample,
saturating the sample, calibrating the transducer, conducting P-S measurement based on
PE or SE method, and determining the residual saturation by an independent means. Note
that in the SE method the nonwetting phase pressure is determined using a mercury
manometer. Therefore, transducer calibration is not required.

The loose soil sample with approximately 2-3% water content was packed in a

brass retaining cylinder (2 ‘/4"OD, 0.065" wall thickness, and 3 cm height) using a
compactor with a 1.6 kg weight falling 10 times from a 7 cm height. This procedure

proved to be effective in maintaining consistent bulk density between samples.

33

Subsequently, the packed soil in the retaining cylinder was assembled with the other
pressure cell parts. The pressure plate was oven dried prior to being installed in the cell.
The ﬁnal cell assembly (Figure 2-1) was placed and connected to the other part of the
measurement device and ready to be saturated.

To saturate the sample, the top of the cell was connected to a vacuum source. The
bottom of the cell was connected to the wetting phase line and the wetting phase pressure
was adjusted to be atmospheric at the center of the cell. Subsequently, in order to
evacuate the soil’s pore gas, a 900 mbar vacuum was applied to the nonwetting phase
without allowing wetting phase to ﬂow into the cell (the valves V10 and V11 were
closed, see Figure 2-2). After several hours, wetting phase absorption was allowed, by
opening either V0 or V11, until the top surface of the soil sample became slightly
oversaturated. At this instance, the suction was dropped to atmospheric pressure. Using
this technique, air bubbles trapped during the advancement of the wetting phase were
conﬁned and collapsed. Next, the cell was removed and weighed in order to obtain what
is refened as the initial weight of the cell. Subsequently, the cell was reconnected to the
wetting and non-wetting phase lines. These lines, including the voids in the transducer,
were ﬁlled with the pertinent ﬂuid and the pressure of each ﬂuid was set to zero. The cell
was then left to equilibrate at this setting for approximately 14 hours after which the
transducer was calibrated.

The transducer is calibrated against a mercury manometer in the following
manner: The wetting phase pressure, either water or oil phase pressure, is ﬁxed at zero.

Once this pressure is set, the cell is removed and the wetting and nonwetting phase lines

 

34

to the cell are sealed. Subsequently, the non-wetting phase pressure is varied stepwise
from zero to maximum and back to zero. The transducer reading then, is calibrated
against the applied nonwetting pressure as indicated by the manometer. This calibration is
conducted prior to any experimental run.

The PE method is employed to determine the P-S relationship in the following
manner. After the transducer is calibrated, the nonwetting phase pressure is increased to
approximately twice the displacement pressure without permitting any desaturation; this
aspect of the procedure will be discussed in the Result and Discussion section. Since no
desaturation has occurred, the increment in the nonwetting phase pressure should not ,
affect the transducer reading even though a small disturbance may be registered. Once the
ﬂuctuation due to this disturbance has diminished, the ﬁrst capillary pressure
measurement is taken. The initial capillary pressure and saturation are therefore known.
The subsequent desaturation is initiated by opening the collector valve. After a ﬁnite
volume has been collected the valve is closed. The collected wetting phase is weighed to
determine the current state of saturation in the sample. The capillary pressure reading is
taken after the transducer reading stabilizes. The procedure is repeated again starting at
the step of opening the collector valve. When the drainage rate becomes too low, the non-
wetting phase pressure is increased. The desaturation process is continued until saturation
in the sample is near residual saturation.

With the SE method, the saturated cell is connected to the wetting phase burette
and the air phase lines are left open to the atmosphere. The pressure in the wetting phase

ﬂuid is adjusted to place the interface in the burette at the center of the cell and the ﬂuids

35

are allowed to equilibrate overnight. The wetting phase pressure at the center of the cell
being atmospheric results in a calculated pressure difference, relative to the air pressure,
of zero. This what is recorded as the capillary pressure at initial saturation.

The next desaturation point is obtained by applying an incremental step of
positive pressure to the non-wetting phase while the outflow is directed to the collector
vial and the line to the burette is closed. Drainage is permitted for at least 22 hours.
Either the mercury manometer or the transducer can be used to determined the
nonwetting phase pressure. However, it is preferable to use the manometer, since it does
not require calibration. The wetting phase pressure is the distance of the wetting phase ‘
ﬂuid surface in the vial to the center of the cell. The saturation in the sample is
determined from the weight of the drained wetting phase. The procedure is repeated for
the subsequent desaturation points. After the last measurement point is obtained, the cell
is weighed. The difference between this weight and the initial weight of the cell reﬂects
the total amount of the wetting phase drained. The agreement between this amount and
the sum of the wetting phase collected in the vial is used to determined the mass balance
error. The cell is then taken apart and all the soil in the retaining cylinder is weighed prior
to being oven dried. The difference in weight before and after oven drying is the residual
saturation and the oven dried weight divided by the brass retainer volume is the bulk
density.

This apparatus was used to examine the performance of the pressure equilibrium
method in measuring air-water and air-oil pressure-saturation data. Likewise, the same

apparatus was used to conduct the saturation equilibrium method measurements of the

36

air-water and air-oil systems. The porous medium used was Metea soil with a texture of
6% non-expansive clay, 11% silt and 83% sand. It contained 1.1% organic matter after
dry sieving to pass 850 micron to exclude larger gravel and other organic matter. In order
to have a stable packed sample, the water-phase used was a deaerated 0.005 M CaSO4
solution as suggested by Klute (1986). The oil-phase was tetrachloroethylene (PCE) at

99.99% purity with a density of 1.623 g/cm3. Building air was used as the air phase.

2. 3. RESULTAND DISCUSSION

2. 3. 1. Pressure-Saturation Measurement Simulation.

The PE method is based on the premise that transient saturation proﬁles within the
soil sample will reach equilibrium relatively quickly when occuning under a zero ﬂux
boundary condition. To verify this notion theoretically, a numerical model was developed
and used to compare the time required by the PE and SE methods to collect P-S data.

This model was a l-D ﬁnite difference model, in which the air mass density and
viscosity are assumed to be negligible. The air pressure and water mass density are
assumed constant with respect to time and space. The appropriate governing equation for

this case is given by Richard equation:

US 3 8h
earn-air, -k,(-5;-l)] {2.1}

where Q is the porosity (cm3/cm3), ks is the saturated hydraulic conductivity (cm/sec), t is

the time (sec), h is the pressure head of water (cm) and z is the vertical distance (cm). The

 

 

37
bottom of the soil sample was set at z=0 cm and the top was at F 3 cm. This height was

divided into 10 uniform elements.

The water saturation (S) and relative permeability (k,) are assumed to follow van

Genuchten (1980) closed-form relations:

1-1/
s = s, + (1 + SJ + (ah)"] " [2-2]
2
-1 /
k. = S,"2[1-é— 53/04))" ) "] [2-3]

where Se is the effective saturation deﬁned as (S- S,)/(l - r). Here Sr, (1 (cm") and n are

the van Genuchten P-S parameters. Note that the model was simpliﬁed by omitting
hysteresis in the P-S relationship and Equation 2-2 was assumed to hold regardless of the
saturation history.

A simulation of the transient conditions within the cell when the PE method is
employed begins at time t, when drainage is initiated by opening the collector valve. The
initial pressure head distribution h(z, t<t,) was hydrostatic and based on the pressure head
speciﬁed at the lower boundary, prior to initiating the drainage. At t, the lower boundary
pressure head is reduced to h,. The ﬁnite difference formulation of Equation 2-1 was
solved to determine pressure head h(z, t), saturation S(z, t) and accumulated outﬂow
volume V(t). To simulate the effect of collector valve closure in the PE measurement,
when the outﬂow volume exceeded the speciﬁed drained volume V, the boundary
condition was switched from h, to a no ﬂow boundary. This was the end of drainage

which occurs at time t,. The execution was continued to simulate transient during

38

saturation redistribution. A PE measurement simulation was executed to determine
transient history in a hypothetical sample. The dimensions used were identical to the
pressure cell as described in the Material and Method section. The parameters describing
the soil-water system used were obtained from air-water P-S measurement using Metea
soil. In the simulation, the sample was initially at hydrostatic equilibrium with lower
boundary pressure head at -35 cmHZO. Pressure head of hp=~50 cmHZO was applied at
the lower boundary to desaturate 0.945 cm3 of water. This scenario is equivalent to
reduce the saturation from 89% to 85%.

Alternatively, the pressure head proﬁle at ﬁnal equilibrium after desaturation can
be determined via mass conservation equation. Calculation based on this equation shows
that; desaturation of Vd=0.945 cm3 reduced the initial pressure head by 2.13 cmH20 at
any point in the modeled domain. The pressure head at the lower boundary, then, would
be reduced from an initiall -35 to -3 7.13 cmHZO by the time the new equilibrium was
reached. Having known this, the SE measurement simulation was conducted by re-
executing the above PE measurement simulation using h,= -37. 13 cmH20. As such, it
was expected that an accumulation of outﬂow as much as V, would only be obtained at
the time when the new equilibrium had been reached. Therefore, in this particular
simulation, the lower boundary condition was never switched to a no ﬂow boundary.

In order to contrast the transient characteristics of the PE and SE measurement
methods Figure 2-3 was plotted. The top plot shows the pressure head at node z=0.15 cm
obtained for the PE and SE simulations. The right vertical axis, labeled as the Error axis,

indicates the percentage discrepancies between the pressure heads obtained by each

 

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39

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40
simulation relative to the equilibrium pressure head at any instance. The equilibrium
pressure head was that obtained via the mass balance equation. The bottom plot in
Figure 2-3 plots the accumulated outflow predicted by the two simulations. The scale of
the right vertical axis in this plots indicates percentage of the target volume V, fulﬁlled
by the accumulated outflow at any given time. These plots shows that, when the PE
method was employed, the pressure head at this node was initially decreased effected by
the imposed lower boundary condition. During this time range, the accumulated outﬂow
increase and reached the target volume V, at t ,= 149 seconds. At this instance, the lower
boundary condition was switched to the no ﬂow boundary condition. This change appears
as constant accumulated outﬂow volume in the bottom plot. The pressure head, as shown
in the top plot, was increased exponentially converging toward the equilibrium pressure
head. At time t larger than 1000 seconds the discrepancies in the pressure head to the
equilibrium head were already insigniﬁcant.

The pressure head curve obtained by the SE measurement simulation shows an
identical trend to the desaturation range in the curve of the PE case where the pressure
head was decreased effected by the imposed lower boundary condition. Differently from '
the PE case, the pressure head curve of the SE case converged directly toward the
equilibrium pressure head since the boundary condition never changed. The accumulated
outﬂow increased slowly. For example, at the time when the PE measurement had

already accumulated 100% of the target volume at t,= 149 seconds, the SE measurement
volume was only capable in accumulating approximately 20% of the target volume. At

time t=1000 seconds, when the PE method was employed, the pressure reading can be

 

41

taken at that time since the error has become insigniﬁcant. When the SE method was
employed instead, at that instance, additional time was still required to accumulate
approximately 50% more outflow volume. Note that, to accumulate this volume the time

required was signiﬁcantly increased. At 10,000 seconds the SE method accumulated only

83% of the target volume.

2. 3.2. Hysteretic Issue in the P-S Measurement Employing the Pressure Equilibrium
Method

The above simulations suggest that the time required to measure P-S data can be
reduced by implementing the PE method. This time reduction, however, would be gained
at the cost of losing accuracy due to the hysteresis problem. To illustrate this problem,
several saturation proﬁles obtained by simulations were plotted. Figure 2-4a plots the
proﬁles obtained by the SE simulation The rightmost curve is the initial saturation proﬁle
S(t,) and the leftmost is the ﬁnal proﬁle S(t,) which is the saturation proﬁle at the end of
the simulation. The four curves between these two proﬁles were the transient proﬁles
during time t,<t<t,. The order of these proﬁles, from right to left, shows the changing of
the saturation proﬁle with respect to time. The position of these proﬁles relative to the
initial proﬁles suggest that, when the SE method was employed, saturation in any
location consistently decreased until the new hydrostatic equilibrium was reached. This
means that the ﬂow was continuously controlled by the desaturation mechanism at any

time during transient throughout the modeled domain. Thus, the data obtained indeed

reﬂected the primary drainage characteristic of the P-S relationship.

42

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Figure 2-4b plots the saturation proﬁles obtained by the PE simulation. Different
from those obtained by the SE simulation, prior to becoming the ﬁnal proﬁle $0,), the
initial proﬁle S(to) changed to the proﬁle at the end of desaturation stage S(td). Therefore,
Figure 2-4b contains two type of transient proﬁles: those obtained during desaturation
transient between time to<t<td and those during redistribution transient between time
td<t<t,. The position of the desaturation transient proﬁles relative to the initial proﬁle
suggest that during time to<t<td the saturation throughout the modeled domain decreased
until the S(td) proﬁle was formed at the end of desaturation stage. After that, at t>td, the
S(td) proﬁle changed to becoming the redistribution transient proﬁle and formed the ﬁnal

proﬁle at the end of simulation time. The relative position of these transient proﬁles
relative to the ﬁnal proﬁle suggest that, in the region of z bellow approximately 1 cm,
saturation was increasing. As a result, the data obtained in the PE case might not

represent the true primary drainage and a bias relative to the SE data would then be

observed.

The hysteresis issue may not be the only determinant which could bias the data
obtained by the PE method. The dynamic state of the fluids at the time when the
measurement readings were taken might also be an important factor. T opp (1967), Smiles
et al. (1971) and Corey and Brooks (1975) observed that P-S measurements taken under
unsteady, steady and hydrostatic states yielded different data sets. They found that at any
given saturation, capillary pressure measured in a steady or unsteady state was higher

than that measured in hydrostatic equilibrium. The model developed cannot be used to

address any discrepancy associated with these hysteretic and dynamic state issues. The

44

hysteresis was omitted in the model so that similarity in the ﬁnal saturation proﬁle S(tc)
obtained by the PE and SE simulations was artiﬁcially imposed. With the model, the
occurrence of the hydrostatic equilibrium state can be precisely determined by monitoring
the change in the pressure distribution proﬁle. During an actual measurement, however,
such a proﬁle might not be practical to be acquired since the height of the pressure cell is
only 3 cm. Thus, any means used to infer the state of hydrostatic equilibrium during the
actual measurement could be less accurate than that employed in the model. In turn, the
P-S point obtained in the actual measurement employing the PE method might be
unintentionally taken while the ﬂuids were still at an unsteady state condition. As the .
model cannot be used to address theses issues, it is necessary to conduct an inspection on

data obtained from experimental measurements.

2. 3. 3. Pressure-Saturation Measurement employing the Pressure Equilibrium and
the Saturation Equilibrium methods.

To verify the PE method experimentally, the method was implemented to measure
P-S data for air-water, and air-oil systems using Metea soil. The PE method would be
justified if it was unbiased relative to the SE method. In order to achieve this condition,
the following criteria should be satisﬁed. First, the measurement should be taken as close
as possible to the hydrostatic equilibrium state. Second, any discrepancies observed
should still be within tolerable limits such that the P-S information obtained is
statistically comparable. This information would include the P—S curve derived from the

measured data and the P-S parameters describing that curve. Subsequent discussions will

be presented from this point of view.

45

232lE'l'i'Zi"!!! 511.5“!

In order to verify the sufﬁciency of the time given for the system to reach
hydrostatic equilibrium for each measurement with the PE method, the transient behavior
of the capillary pressure as indicated by the transducer during redistribution was recorded
approximately every 5 minutes. Typical plots of such data are shown in Figure 2-5. The
open squares in Figures 2-5a through 2-5d denote the capillary pressure readings. The
series of points belonging to a particular desaturation step end with a solid point for
which hydrostatic equilibrium was assumed to be achieved. The criterion for identifying
these end points was that the rate of capillary pressure change be less than 0.25 mbar per
minute. If this rate was achieved at the second point, additional readings were taken. The
label next to each solid point shows the degree of saturation at that step. At S = 0.6T and
0.25, the pressures were equilibrated overnight (785 minutes). The horizontal lines

indicate the pressure level at which the non-wetting phase pressure was set (labeled as hp
in the Figure 2-5).

Having knowledge of the trend of the pressure head proﬁle in the vicinity of the
pressure plate as shown in Figure 2-3, it was possible to reconstruct the redistribution
curve of each desaturation step. Each redistribution curve in Figure 2-5 (solid line) was
traced manually through the open points that belong to a particular step ending at the
solid point. The beginning and the end points of this solid curve were then manually
extrapolated (shown as dotted lines) so that at the beginning of each desaturation step it
crossed hp and at the end point it continued the trend of the solid line. It is assumed that

the quasi-equilibrium state was already reached at the points where the redistribution

46

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47
curves became horizontal. Beyond these points, the rate of the capillary pressure change
was assumed to be insigniﬁcant. It was observed that the solid points were laid in the
horizontal region of the curve. This indicates that the criteria of 0.25 mbar per minute was
sufﬁcient to assure the quasi-equilibrium state of the system. Inspection of the
constructed curve indicates that, with the air pressure setting hp as shown in Figure 2-5, in
an air-water system with high saturation (980%), 20 minutes was sufﬁcient to reach
quasi-equilibrium. At moderate saturation (30%<S <80%), it was 40 minutes and 60
minutes at lower saturation. In a similar manner of analysis in an air-oil system, those

values were approximately 15, 30 and 45 minutes respectively.

cum”: 1"- ..A' 'Itl’tl 1’ ’ .' I. 1.1141! 1’

S . E T! . l I E I
Comparison of the P-S curVes obtained by the PE and SE method is presented in
Figure 2-6a and 2—6b. The solid curves in these ﬁgures are the best ﬁt curve of Equation
2-2 for the P-S data measured using the SE method, pooled ﬁ'om 5 replications of
air-water measurements (Figure 2-6a) and 4 replications of air-oil measurements (Figure
2-6b). The dashed lines were similar to the solid curves but determined from data
obtained by the PE measurement. In this case, the obtained P-S curves for the air-water
and air-oil systems were based on pooled data from 7 and 4 replications respectively.

The shaded areas are regions within the 95% conﬁdence limits of the solid curves

which were determined based on Seber and Wild (1988):

8S=t4f ste [1+f;(FT F)—l fh]

 

48

The term (‘21; is the student t value at 95% conﬁdence intervals with degree of ﬁ'eedom df,

and sse is the sum of the squared error. Since Equation 2-2 was used to describe the P-S

relationship, the covariance matrix F is deﬁned as:
[F]: [35011) 3305') 35070]
3Sr 3a an
where i is the data index number. The matrix 1' is the covariance matrix at a given
pressure.

Prior to comparing the P-S curves obtained by the methods, the characteristic of
the curve obtained by the reference method, i.e. the SE method, will be discussed. The
solid lines in Figure 2-6 are assumed as the best estimate of the population primary
desaturation curves of the air-water and air-oil systems when measured in the Metea soil.
The conﬁdence limits of this curves bound an area in which, within 95% probability, a
different best estimate curve might be laid if a different data set from the same population
were used. At a given pressure, the width of the conﬁdence limits is dictated by the
variability of the saturation data across sample replications. In general, this variability
might be due to the deference from one replica to the next as to their: 1) applied pressure,
2) equilibrium state when the readings were taken, 3) pore size distribution and 4)
precision in determining the saturation. In conducting the SE experiment, ﬁve
replications were run simultaneously. A single air pressure source was regulated and
distributed to each replica using a manifold. Therefore, it is assumed that each replica

received identical load of air pressure in each desaturation step. As mentioned in the

Materials and Methods section of this chapter, with the SE method the samples are

49

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51
permitted to equilibrate for at least 22 hours. Subsequently, the saturation in each replica
was determined one at a time. The time different between the ﬁrst and the last replica was
not more than 5 minutes. It is unlikely that this difference contributed a signiﬁcant effect.
If we assumed that 22 hours given in each desaturation step was sufficient to equilibrate
the sample, then the width of the conﬁdence limits were affected only by the variation in
the pore size distribution across the replication and the precision in determining the
saturation. Across the replication, the variation in the pore size distribution is due to the

inability to produce the same pore geometry from one sample to the next.

To discriminate the effect of these two determinants on the obtained conﬁdence
limits the saturation measurement precision was inspected. The discrepancy between the
total mass of the wetting-phase released by the soil sample to that accumulated in the
collector vial in each replica was determined. This discrepancy is the total saturation loss
in one replica. It is true that the ﬁ'action of this loss that occurs in each desaturation step
was unknown. It is assumed that this provides a conservative reasonable etimate of the
error that could occur at any step. Out of 5 replications of air-water measurements the
maximum saturation error in the worst replica was only 0.009. The plot in Figure 2-6, on
the other hand, shows that the narrowest band width of the conﬁdence limits in the
air-water curve was 0.184. For the case of air-oil the maximum measurement error from 4

replications was only 0.017 compare to 0.105 of the narrowest conﬁdence limits. It
appear that the maximum measurement errors were small when compared to the

conﬁdence limits band width. Thus, the conﬁdence limits of the P-S curve obtained via

52

the SE method was mostly affected by the pore size distribution variation across the

replication rather than the measurement error.

As mention previously, the PE method could be justiﬁed if it is capable of
producing data so that any P—S information derived from these data are as good as those
determined from data measured via the SE method. It is observed in Figures 2-6a and 2-
6b that the discrepancies in the P-S curves estimated by the two methods are small
compared to the band width of the conﬁdence limits of the reference curve. For example,

in the case of the air-water system, the highest discrepancy between the two curves was
only 0.03 which occurs at capillary pressure h=800 cmHZO. On the other hand, the width
of the conﬁdence intervals of the SE curve at this pressure was 0.230. In the case of the
air-oil system, the highest discrepancy was 0.021 at h=30 cmHzO while the conﬁdence
limits width was 0.110. As stated before, the conﬁdence limits deﬁne the area in which,
within 95% probability, any best estimate curve from the same population might be laid.
Therefore, small discrepancies relative to the conﬁdence limits band width, such as those
observed in the investigated cases, suggest that the obtained P-S curves via the PE and SE
methods were derived from two data set which came from the same population. As such,
the data obtained by the two methods were not deferent statistically.

Additionally, the relative position of the P-S curve obtained via the PE method to
that by the SE method, was not indicative of any consistent bias. Any bias associated with
the hysteretic issue would be observed as a departure with consistent direction from the
reference curve. The effect of hysteresis would shift the curve toward the primary

imbibition scanning path. Thus, in Figure 26, it would appear as a consistent departure

53

toward the left side of the reference curve. Such a consistent tendency was not observed
in any of the above cases investigated. It is speculated that the effect of hysteresis, if any,
was overwhelmed by the variation due to sample replication. This suggests that, at least

for the investigated case, the effect of hysteresis is insigniﬁcant compared to the pore size

distribution variation across replication.

In!!! It! 1’ '- '4 4/1” I: (1’?! 1’ ' ’ .’ .€.I .1141!

15] 'E'l'l'llll

Typically, the purpose of a P—S measurement is to obtain information to determine

the P-S relationship. In most cases, this information is required in the form of a P-S
parameters set. Note that the comparison plots in Figure 2-6 reveal only the effect of
parameters discrepancies when they are used to construct a P-S curve. A different use,
such as to determine unsaturated hydraulic conductivity, might result in a different level
of agreement. However, if those parameters came ﬁ'om the same population, any
disagreement associated between the discrepancy of the PE and SE methods could be
regarded as just variation within one SE measurement to the next. Therefore, it is
necessary to inspect whether the P-S parameters determined ﬁ'om the obtained PE data
are comparable to those obtained by the SE method. The following paragraphs discuss
this issue.

The P-S parameters used to construct the P-S curve in Figure 2-6 are listed in

Table 2-1 along with their 95% conﬁdence limits. These limits were determined

following Seber and Wild (1988):

54

 

 

Esr
’SSE
ea = ’35 . a; .[C]
-en.

where 85,, 8a, and 8,, are the 95% conﬁdence limits of parameters S,, a and n respectively
and [C] is the diagonal of the covariance matrix (FT.F)".

Table 2-1: The best estimated air-water and air-oil
P-S parameters based on the pooled data
measured using the PE and SE methods

 

 

 

SE Method PE Method
Air-Water
S, 0.193zt0.059 0.22%0058
n 2.91i0.68 3.12:1:058
a 0.0213i0.0032 0.0218zt0.0014
Replication 5 7
Air-Oil
Sr 0.231i0.047 0.257:l:0.058
n 2.50:1:0.38 2.80i0.83
a 0.0646i0.0081 0.0637dc0.0101
Replication 4 4

 

Following the current approach, the parameters determined from data measured
using the SE method are assumed to represent the characteristic of the population’s
parameters. As shown in Table 2-1, in all cases, the parameters estimated from data
measured via the PE method are within the conﬁdence limits obtained in the case of SE
method. It is also observed that the width of the conﬁdence limits of the parameters in the
PE case are comparable to those in the SE case. This indicates that the parameters from

the PE and SE cases share similar statistical properties. None of the investigated cases

55

suggests that the parameter set estimated from the PE data is different from those
estimated from the SE data. Based on this, it is concluded that the obtained parameters
estimated from data measured via the PE and SE method are comparable. In turn, the PE

measurement method is capable of providing data as good as the SE method.

2. 4. SW YAND CONCLUSION

An alternative method to reduce the P-S measurement time based on the pressure
equilibrium (PE) method developed by Su and Brooks (1980) was investigated.
Theoretical veriﬁcation using a mathematical model conﬁrmed the feasibility of this
method. To justify its acceptability, the performance of this method was compared to the
conventional saturation equilibrium (SE) method. In order to conduct this comparison,
the Su and Brooks (1980) apparatus was modiﬁed and used to measure air-oil and
air-water desaturation data employing the PE and SE method. The soil used was Metea
soil with loamy sand texture and the oil phase was tetrachloroethylene.

A mathematical model was developed to simulate the transient ﬂow during P-S
measurement using either the PE or SE method. The simulated measurements suggest
that the time required to measure P-S data can be reduced by implementing the PE
method. The simulation indicated that the PE method was able to reduce the saturation
from 89% to 85% within 149 seconds. During this time only 20% of the target volume
was accumulated by the SE method. At time t= 1000 seconds, the pressure head error in

measurement employing the PE method was already insigniﬁcant while in the SE method

56

additional time was still required to accumulate approximately 5 0% more outflow
volume.

This time reduction, however, would be gained at the cost of losing accuracy due
to hysteresis. The simulation suggests that during the redistribution stage in the PE
method the saturation in the lower region of the sample was increased. As a result, the
data obtained in the PE case might not represent the true primary drainage and a bias
relative to the SE data would then be observed.

Comparison of the P-S curves obtained by the PE and SE method, however, did
not show any indication of bias. It is speculated that the effect of hysteresis, if any, was.
overwhelmed by the variation due to sample replication. This suggests that, at least for
this uncosolidated soil, the effect of hysteresis was insigniﬁcant compared to the pore size
distribution variation across replication.

None of the investigated cases suggests that the parameters estimated from the PE
data are signiﬁcantly different from those estimated from the SE data. Based on this, it is
concluded that the obtained parameters estimated from data measured via the PE and SE
method were comparable. In turn, the PE measurement method is capable of providing
data as good as the SE method.

The effective time required to measure 19 desaturation data points by the PE
method was only about 12 hours. Compared to the 108 hours required by the
conventional method, this was a signiﬁcant improvement. This time reduction was
theoretically conﬁrmed. Implementation of this method to determined the air-water and
air-oil P-S relationship in loamy sand Metea soil demonstrate the capability of this

method to substitute the conventional method. Based on this, it is concluded that the PE

57

method provides a more efﬁcient alternative for measuring P-S data.

CHAPTER III

T HE ACCEPTABILITY 0F SCALING THEORY IMPLEAIENTA TION IN
PREDICTYNG THREE-PHASE CONS” TU 17 VE E QUA T IONS FROM
TWO-PHASE PRESS URE-SA TURA 17 ON DA TA.

3. 1. BACKGROUND

Recently, Parker, et al. (1987) proposed a predictive method employing scaling

theory to construct three-phase constitutive information relating relative permeability (kr),

saturation (S) and capillary pressure (P). The predictive method (Leverette, 1941)
assumes that when air, oil and water phases coexist in soil pore, the water saturation is a
function of oil-water capillary pressure only and the sum of water and oil saturation is a
function of air-oil capillary pressure. Therefore, the three phase P-S relationship

information can be obtained from two-phase P-S measurements. Once the three-phase

P—S relationship is known, the three-phase kr-S relationship can be determined using

various formulations.

Simpliﬁcation through the predictive method is widely utilized (Corey, et al.,
1956; Stone, 1973; Azis and Settari, 1979; Parker, et al., 1987) primarily due to difficulty
in directly measuring the capillary pressure of an individual phase in a three-phase system
(A briola, 1989). Even in utilizing the predictive methods, the experimental burden

remains substantial. For this reason, Parker, et al. (1987) proposed a predictive method

employing scaling theory. In this method, a complete set of two- and three-phase kr-S-P

58

59
equations for the desaturation scanning path for a particular porous medium were

formulated as dependent solely upon the P-S desaturation parameters for a single two-
phase system and scaling factors. When the scaling factors are predicted from interfacial
tension measurements, the experimental burden is reduced substantially.

This scaling theory is based on the premise that in a porous medium there exists a
scaling factor ([3) which transforms the P-S curve from a particular two-phase system into
a different two-phase system (Leverette, 1941; Parcel, 1942; Miller and Miller, 1955).
Parker et al. (1987) further postulated that B can be predicted from the ratio of the
interfacial tensions (0') of the immiscible ﬂuids involved. Lenhard and Parker (1987)
validated this notion by comparing the scaled two-phase P-S curves to those obtained
from direct measurements. The incorporation of scaling theory into the predictive method
was validated by Lenhard and Parker (1988). This validation was conducted by
comparing the water and total liquid P-S data measured in three-phase systems to those
constructed via the scaling method. These validation studies support the use of scaling
theory.

The kr-S-P relationships constructed via the predictive method employing scaling

theory are based on a rigid porous medium assumption. This assumption states that the

soil pore structure is constant with respect to time. According to Parker, et al. (1987), the

prediction of kr-S relationships are sensitive to any disturbance in pore size distribution

such as that due to swelling. Rather small changes in pore size distribution, which may

not be clearly evident in P-S relations, may markedly affect the permeability functions. If

60

this notion is true, it might have signiﬁcant impact on the implementation of the scaling
theory. Note that, the characteristics of the soil pore structure is constituted in the P-S
parameters. To obtain these parameters, the scaling theory relies on data that are

measured in one system only. Therefore, any small error in these parameters might
signiﬁcantly biases the prediction of the kr—S-P relationships. From this point of view, the

accuracy and precision of the P-S parameters measured in that single two-phase system
appear to be important. From here on, these parameters are referred to as base parameters.
The accuracy and precision of the P-S parameters are affected by the soil texture
involved (McCuen er al. , 1981). The parameters determined in a soil containing fractions
of sand, silt and clay tend to give less variation than those determined in uniform grain
size soil. The accuracy and precision might also vary from one immiscible ﬂuid system to
another. For example, displacement pressure measured by Demand and Roberts (1991)
possesses higher variations in air-water system than in oil-water system. Lenhard and
Parker (1987) observed that their P-S measurements involving benzyl alcohol were
inferior compared to other systems they had investigated. They concluded that this was
due to the characteristics of benzyl-alcohol which drains at a relatively low capillary

pressure.

The variances of the base parameters might be different when measured in

different soil textures or different ﬂuid pairs. In turn, the kr-S-P relationship predicted

based on these parameters might also possesses different accuracy and precision. This

chapter describes the investigation of the implementation of scaling theory in the

61

predictive method to predict the kr-S-P relationships when either air-water or air-oil

systems were used as the base system. These two approaches, ie. using air-water or air-oil
in implementing the scaling theory, are referred to as method 2 and method 3,
respectively. The obtained relationships were compared to those obtained via the
predictive method without utilizing scaling theory. The latter approach served as the
reference method and was referred to as method 1. These comparisons were conducted
utilizing two soils that have a contrast in their textures; loamy sand Metea soil and
uniform grain Ottawa sand.

Using either the air-water or air-oil systems as the base system has an advantage
over the use of an oil-water system. Air-water P-S information is considered as the most
common data available compared to the other two systems. Also a well established
measurement protocol such as that by Klute (1986) is available. When lacking air-water

data, an air-oil system is more attractive for two reasons. First, the air-oil P-S parameters
are directly involved in the three-phase kr-S-P equations. Secondly, compared to the

oil-water system, the air-oil measurement technique is more similar to the well
established air-water P-S measurement.

The scaling theory has been incorporated in a versatile numerical three-phase ﬂow
model (Kaluarachchi et al., 1990). This model was improved in becoming valid for both
the desaturation and imbibition scanning paths (Parker and Lenhard, 1987; Lenhard and
Parker, 1987; Lenhard et al., 1989) and accepting multi-component organic chemical

transport (Parker and Kaluarachchi, 1990). This model is one among a few that accepts

62
dense non-aqueous phase liquids (DNAPL) as the oil phase (Huling and Weaver, 1991).
Unfortunately, most of the scaling theory studies use light non-aqueous phase liquids
(LNAPL) as their oil phase. The only instances of DNAPL uses were tricholoroethylene
(Parker and Lenhard, 1987a) and benzyl alcohol (Lenhard and Parker, 1987). However,
the density of benzyl alcohol is only 1.046 g/cc which is very close to the density
threshold of the LNAPL type of oil ( < 1.00 g/cc). Thus, the small contrast offered by
benzyl alcohol may not be sufﬁcient to reveal the characteristic of DNAPL. Based on
this, tetrachloroethylene, which ha denity of 1.6 g/cc, was selected in this study as the oil

phase to broaden the type of dense NAPL involved in the scaling theory study.

3.2; T HEORE ”CAL BACKGROUND.

Parker et al., (1987) described a predictive method to construct the three-phase

kr-S-P relationships from P-S data of the two-phase system. The subsequent paragraphs

summarize the theory in conjunction with its application and focus on monotonic
desaturation involving DNAPL in the oil phase.

The scaling factor (B) was deﬁned as a constant which transforms the P-S curve of
a particular two-phase system into a different two-phase system in a single porous
medium. Speciﬁcally, this constant relates the capillary pressure of two two-phase
systems at a given saturation. The saturation which measures the fraction of soil pore
occupied by the nonwetting-phase, is dictated by the interfacial surface and pore

geometry. In turn, identical saturation implies identical pore geometry and radius of

63

curvature (RC) of the interfacial surface. Employing the Laplace theory, the ratio of the

capillary pressures of two two-phase systems at a given Rc is equal to the inverse of the

interfacial tension ratio. If air-water is chosen as the reference system, that statement can

be formulated as follows:

J— —331 [3-1]

B
WEthRcopq

In which the hpq are the capillary pressures of any arbitrary two-phase system pq (=aw,

ow, ao) expressed in terms of water capillary head.
The scaling factor can also be expressed in the term of the P-S ﬁmctional

relationship equation. van Genuchten (1981) formulated the P—S functional relationship as

follows:

l/nm
55%th )=Sr-pq +(1 Sr-m)[l+@pq Hmym] [3‘2]

This equation is referred to as the P-S model. The constants S,_pq,a pq, and nm are the

curve ﬁtting parameters. Employing this equation, the ratio of haw and hpq becomes:

Z:=[“—1’E§Zi:::::§

At the same saturation and assuming S,.aw=S,.pq and naw=npq for identical porous

aaw

medium, the second bracket on the right hand side is unity. This derivation leads to an

alternative way of formulating the scaling factor, which is:

- "W = a—Pq- [3-3a]

The aaw and am can be obtained by regressing the pertinent P-S data into the VG model.
Combining Equation 3-1 and 3-3a leads to:

0
“pg 2 quaaw "‘5 3231a“, [3’3b]
pq

Incorporating Equation 3-3b into Equation 3-2 leads to:

n 1/ n-l
O’
qu(qu,hpq)=S,+(1-S, ”(aw-flit”) [34]
P9
This equation permits prediction of the P-S curve of the system pq from air-water

parameters. Note that, in incorporating Equation 3-2 into the scaling theory the parameter
Sr was assumed to be independent of ﬂuid properties or saturation history. The parameter
n was also formulated without any designation of the ﬂuid system. Several studies, such
as those by van Genuchten (1980), van Genuchten and Nielsen (1985) and Parker and
Lenhard (1989), showed that n is equivalent to the parameter 7t in the Brooks and
Corey (1964) P-S model. The parameter A. was postulated as an empirical parameter used
as the index of pore size distribution (Carey, 1986). In turn, n was also regarded as a

porous medium-dependent parameter.

Following the notion in Equation 3-2 and Equation 34, the prediction of oil-water

and air-oil systems from an air-water system can be expressed in the following manner:

83°(hao)=53°(8.o.h.o) [3-5]
53‘" (how)3 531‘" (ﬂowahow) [3‘6]

65

The leﬁ hand side reﬂects the direct measurement, while the right hand side expresses the
prediction using the scaling theory. The predictions of the three-phase P-S relationship

from the two-phase P-S information were given by:

53°“ 253% [3-7a1
=SS°(h.o) [3-7b]
53°“ = SS.“ (how) [3-81
Sgow = Staow _ 330w [3 _9a]
=83°(hao)- SSW...) [3-9b]

The subscript t denotes the total saturation of the oil plus water phases in the three-phase
system. The superscripts indicate the pertinent phase systems, either a three-phase (aow)
or a two-phase (ao, ow) system.

Once the three-phase P-S relationships were obtained, the k-S relationships can be

constructed. Employing the VG parameters, such a procedure was formulated so that:
2
m _m /2 m In " m m "
k;°*=(§. —S.. {[1—6. ) ] —[1—(§. )H [3-10]
2
now /2 aow In In
Ma ){i-[i-(. )1} [3.11]

where m=1-l/n, 1:33“ and k3,?“ denote the relative permeability of the water and the oil
phase in a three-phase system receptively. Omitting the superscript, the effective
saturation Sq is deﬁned as (Sq- l.)/(1- r).

Note that, a three-phase system with Stww = 1 is equivalent to a two-phase oil-

water system. In this particular case, when the oil and water phases are regarded just as

66

the nonwetting and the wetting phases of a two-phase system, Equation 3-10 becomes the

equation for the nonwetting phase relative permeability kg]. In this two-phase form the

saturation S3” becomes saturation of the wetting phase 85" . Likewise, at S3” = 0 any
three-phase system becomes a two-phase air-oil system. In this case the oil phase act as
the wetting phase and since Sta” contains only the oil phase it becomes saturation of the

wetting phase S 5" . In this situation Equation 3-10 becomes the equation for the wetting
phase relative permeability kg! . Thus, the two-phase relative penneabilities kg , k3," ,

kfow and k3: can be calculated using Equation 3-10 by substituting appropriate saturation.
Previous investigators focused their studies by testing the hypotheses as
formulated in Equations. 3-5, 3-6, 3-7b, 3-8 and 3-9b. Lenhard and Parker (1987) studied
the agreement of the mean response of the direct measurements (the left hand side of
Equation 3-5 and Equation 3-6) and the predicted ones (the right hand side of Equations
3-5 and 3-6). Later, Lenhard and Parker (1988) focused their study on the validity of
Equations 3-7b, 3-8 and 3-9b. The favorable agreements they observed deteriorated

slightly when the right hand side of those equations was substituted in Equation 3-5 and

Equation 3-6. Nevertheless, they showed the possibilities of constructing three-phase k,-

S-P relationships when a set of parameters Sr, n, (row and Ciao is given. By predicting the
scaling factor from interfacial tension and choosing the air-water as the reference system

the required parameters are 5,, n, Bow, B80 and (13“,.

67
3.3. MA 75121.41. AND Mzmoo.

In order to obtain the parameters required to construct the kr-S-P relationship

using methods 1, 2 and 3 which were discussed previously, P-S measurements of
air-water, oil-water and air-oil systems were conducted using Ottawa sand and Metea

soil. Ottawa sand is a relatively uniform sand. About 85% of its grain diameter is 0.33
mm and no grains are smaller than 50 u. Metea soil has a loamy sand texture with 6%
being non—expansive clay, 11% silt and 83% sand. It is a structureless soil that contains
1.1% organic matter after it is dried and sieved to pass 850 u. The water-phase used was a
0.005 M CaSO4 solution as suggested by Klute (1986). Prior to being used in the
experiment, this solution was subjected to a 750 mbar vacuum for at least two hours in
order to minimize the amount of dissolved air. Tetrachloroethylene (PCE) at 99.99%
purity and density of 1.623 gr/cm3 was used as the oil-phase.

The P-S measurement protocol was based on the pressure equilibrium method as
described in more detail in the Materials and Methods section of Chapter II.
Conventionally, the P-S measurement in a pressure cell is conducted by measuring the
saturation at its equilibrium state under a certain imposed capillary pressure. The pressure
equilibrium method, which is based on the method proposed by White et al., (1970b) and
Su and Brooks (1980), reverses the conventional measurement method. In this method the
equilibrium state of the capillary pressure is measured under a series of desaturation steps

which are externally controlled. By employing this method, large numbers of P-S data

can be obtained in a relatively short time.

68

The mentioned protocol was applied to measure the P-S relationship in a soil
sample which was placed in a pressure cell. The pressure cells used were the Tempe cells
by Soil Moisture type no. 1400 made from Plexiglas. All standard rubber o-rings of the
cells were replaced with Viton o-rings to avoid dissolution and to minimize PCE
absorption. A l-bar pressure plate made by Soil Moisture (part no. 1430B1M3) was used
as the nonwetting-phase barrier. The loose soil sample was placed in a brass retaining
cylinder (2%"0D, 0.065" wall thickness and 3 cm height) and packed using a 1.6 kg.
falling weight dropped 10 times from 7 cm high. With this method, replicable bulk
density of i1 .67 gr/cm'3 for Ottawa sand and 21:1.44 gr/cm'3 for Metea soil was achieved.
The packed samples in the cells were vacuum saturated and were equilibrated at zero
capillary pressure for at least 12 hours. A pressure transducer, which had been calibrated
against a mercury manometer prior to any measurement, was used to monitor the
capillary pressure. Measurements of each system were replicated four to eight times. Out
of these, the three replicates that yielded the least mass balance error were selected to be
used for further analyses.

The P-S parameters used in method 1 were detemrined by regressing the VG

equation into the measured P-S data. The regression procedure was constrained to yield
common values of S, and n in all two-phase systems from the same soil type. This was
necessary to accommodate the assumption which states that S, and n are porous medium-

dependent parameters. To achieve this, the sum of the squared error in predicting the

air-water, oil-water and air-oil parameters was minimized simultaneously (pooled

69

regression). The parameters S,, n and (raw required by method 2 were obtained from

regression upon air-water P-S data only (individual regression). The cow and aao were
determined via Equation 3-3b. For method 3, the required parameters were obtained in a
similar manner but the air-water system was replaced by the air-oil system.

The precision limits of method 1 for Saw , Saw and Sf,” were the 95% conﬁdence
limits of the regression determined using the procedure as explained by Seber (1988). For
the other constitutive variables, those limits were determined by propagating the 95%
conﬁdence limits of the parameters involved. The procedure employed was based on
error propagation method utilizing the ﬁrst order of Taylor series expansion. An identical
propagation method was also used to determine prediction precision by methods 2 and 3.

To obtain the interfacial tension data required by methods 2 and 3, the air-water

oil-water and air-oil interfacial tensions were measured using the pendant drop method by

Adamson (1976). Values of cow were measured at 20 0C using oil and water taken from
the air-oil and oil-water P-S experiments. The ﬂuid samples used to determined an and

03“, were taken from the air-oil and the air-water P-S experiments. As such, the liquid

had been put in contact with the porous medium material for about 60 hours.

3. 4. RESUL T AND DISCUSSION

To verify the acceptability of method 2 or 3 relative to method 1, each method

was implemented to construct the k,-S-P relationships. The P-S and interfacial tension

70

data were measured. Subsequently, the P-S parameters and scaling factor values including

their precision limits were determined. Table 3-1 lists the results obtained both for
Ottawa sand and Metea soil. Based on these values the constitutive variables S‘an (h),
Saw (h), Saao (h), 33°“ (h), k3 (h), kg," (h), k2: (h) and k3,?“ (h) were plotted and

superimposed with their uncertainty limits (Figure 3-1 through Figure 3-9).

Table 3-1: The obtained interfacial tensions, P-S parameters and
scaling factors '

 

 

Ottawa sand Metea soil

Interfacial terrsion
O'aw (dyne/c 71.60 $1.661 69.09 $5.512
cow (dyne/c 38.03 $3.001 38.62 $4.591
oao (dyne/cm ) 26.40 $1.762 25.29 $1.650
Method 1
S, 1 0.11 $0.048 0.22 $0.036
aaw (cm'l) 0.053 $0.0037 0.023 $0.0015
(tow (cm; ) 0.093 $0.006] 0.045 $0.003]
am (cm' ) 0.134 $0.0322 0.058 $0.0040
n 4.33 $0.741 3.10 $0.352
Method 2
S, 1 0.08 $0.075 0.21 $0.065
aaw (cm' 1) 0.053 $0.0045b 0.022 $0.0016
crow (cm; ) 0.099 $0.01 17b 0.040 $0.0064b
orao (cm' ) 0.142 $0.0158 0.060 $0.0067
n 4.13 $1.116b 3.38 $0.775!)
Bow 1.88 $0.155b 1.79 $0.256b
Ba0 2.71 $0.192 2.73 $0.281
Method 3
S, 1 0.14 $0.077 b 0.23 $0.068 b
“aw (cm' 1) 0.037 $0.005 l b 0.023 $0.0033
(10w (cm; ) 0.068 $0.0108 0.041 $0.0069
“a0 (cm' ) 0.099 $0.01 18 0.063 $0.0062
n 3.00 $0.728b 2.882 $0.540b
38w 0.37 $0.026b 0.37 $0.038b

0.69 $0.072 0.65 $0.089

 

') Except where indicated, all values were given as mean $95%

conﬁdence limit from parameter estimation procedure.
b) The values given as mean $ uncertainty from error propagation

employing the ﬁrst order of the Taylor series.

71

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

so
60'
g:
35‘
30‘
A A
g 25' g
5 E.
-= _ -:
20
15‘
10‘
5.
o . , r .
.01 0.0 oz 0.4 0.6 0.8 1.0 12
SW
(a) Ottawa Sand: Oil-Water
2.50
250‘
225 200‘
f I
150’
120:, no"
no" 100‘
100‘ 90'
90‘ so‘
A _ A
a 80 70'
D -
5 7° 5 GM
4
-: so‘ -= d
504
so'
_ ‘0‘
40
- 30‘
3o .
20' 2°-
10 10-
o l o I I
.02 0.0 —02 0.0 02 0.4 0.6 0.8 1.0 12
w 33°
(c) Metea Soil: Oil-Water (d) Metea Soil: Air-Oil

Figure 3-1: Two-phase oil-water and air-oil pressure-saturation curves obtained by

method 1 (solid lines) with their 95% conﬁdence limis (shaded area)
superimposed with those obtained by method 2 (heavy dashed lines) with
their uncertainty limits (light dashed lines) and the measured data.

M24234:

72

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E
.0
E
v:
00.0 02 0.4 0.6 0.3 1.0 12 0.0 . 02 0:4 ' 0.6 ' 0.0 1.0 1.2
553' S3."
(3) Ottawa Sand: Air-Water (b) Ottawa Sand: Oil-Water
600 250
500 .
..., — ..., h
f 120‘
225 ' 110
200‘ 10°.
. 90,
115 q‘
80 1
E 150 ' ’5 70:
“D 125 ' "D ‘
E 8 6°
.: 100' I: 50-
- 4° '
7s .
30 ‘
50 ‘ _
. 20
25 10‘
0 o
0.0 02 0.4 0.6 0.0 1.0 1.2 0.0
Si‘v” 8%"
(c) Metea Soil: Air-Water (d) Metea Soil: Oil-Water

Figure 3-2: TWo-phase air-water and oil-water pressm'e-saturation curves obtained by
method 1 (solid lines) with their 95% conﬁdence limis (shaded area)
superimposed with those obtained by method 3 (heavy dashed lines) with
their uncertainty limits (light dashed lines) and the measured data.

Ian-treat

73

 

 

'o
.....
.
c
.

      

  

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a _ g -
a; 01 1 E 01
l (3) Ottawa Sand (b) Metea Soil
Air-Water Air-Water
0.01 1: 0.01 4.}
h (mbar)
1 1 10
1'. 11H ‘ ‘ ‘ ...... ..
l
3 .
53. 0'1': of 01 3
(d) Metea Soil
Oil-Water
0.01 0.01 it
h bar
on ) 10 1
l 1 1 1
8:. or - 8:. 01 -
0.01 0.01 t;

Figure 3-3: TWO-phase air-water, oil-water and air-oil relative permeability curves
obtained by method 1 (solid lines) with their uncertainty limits (shaded
area) superimposed with those obtained by method 2 (heavy dashed
lines) with their uncertainty limits (light dashed lines).

n-l2.cdr

74

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

       

3 3
“,3- 01 “.3- 013
0.01 0.01
1
1': l1
4
3
53- 01-: of 01
(d) Metea Soil
Oil-Water
0.01 0.01 'l
h (mbar)
1 10
1‘1 .............. 1‘.
4
8 8
.5 0.11 E 0.11
(e) Ottawa Sand
Air-Oil
0.01 0.01 {I

 

 

 

 

 

 

Figure 3-4: Two-phase air-water, oil-water and air-oil relative permeability curves
obtained by method 1 (solid lines) with their uncertainty limits (shaded
area) superimposed with those obtained by method 3 (heavy dashed lines)
with their uncertainty limits (light dashed lines).

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(b) Metea Soil

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3-7: Three-phase oil relative permeability obtained by method 1 (solid
lines) and method 2 (dashed lines).

13.12.34“

80W
km

0.01

kgw

  

: (a) Ottawa Sand

 

ha0 (mbar)

10

how= 100 mbar
Sw= 0.00

 

 

 

how (mbar)
1 0

 

(c) Ottawa Sand
h = 4 mbar
s?- 0.95

 

 

 

 

0.01

78

kg“

 

 

 

~
'n
-
~
"
u

   

(b) Metea Soil
= 100 mbar

0W

sw= 0.05

 

 

 

0.01 ‘

how (mbar)
_ .100

.--
U...
,-
a
,-
4

    

(d) Metea Soil
h = 4 mbar

5? 0.99

 

 

Figure 3-8: Three-phase oil relative permeability obtained by method 1 (solid
lines) with their uncertainty limits (shaded area) and those obtained
by method 2 (heavy dashed lines) with their uncertainty limits (light

dashed lines).

I342”

1.3:“

0.01

(a) Ottawa Sand
h = 100 mbar

0W

sw= 0.00

 

10

     

 

 

    

 

0.01

(c) Ottawa Sand
f hm: 4 mbar
s,= 0.95

.......
.....
,-
_o
,-
.0
.
o
.0
r

 

 

 

79

0.1 1

k2,?“

(b) Metea Soil

 

hm (mbar)

10

 

......
......
..-
.....
.-
.
c
u
m
u
s
m
a
-

 

h = 100 mbar
sf} 0.04

   

 

 

0.01

 

0.1 1

it :

 

0.01 "

 

 

 

Figure 3-9: Three-phase oil relative permeability obtained by method 1 (solid
lines) with their uncertainty limits (shaded area) and those obtained
by method 3 (heavy dashed lines) with their uncertainty limits (light

dashed lines).

til-13nd:

80

In method 2, the base parameters were used to construct the curves of water

saturation Saw against capillary pressure h for the air-water system using Equation 3-2.
The 95% conﬁdence limits obtained during regression were used as the uncertainty limits

of the curve prediction. The same base parameters plus the measured interfacial tension

were used to plot the air-oil Sf,lo and oil-water Saw curves using Equation 3-4. The
uncertainty limits of these curves were determined based on the uncertainty in the base
parameters and the scaling factors. The uncertainty limits of the base parameters were
assumed to be identical to their conﬁdence limits obtained in the parameter estimation
procedure. The uncertainty limits of the scaling factors were determined by propagating

the conﬁdence limits of the interfacial tension values.

The two-phase relative permeability curves k3: k3: and k3,},v were plotted

employing the two-phase form of Equation 3-10. With this two-phase form, the terms of

Stww and S3," in Equation 3-10 were substituted by SEW , Saw or 53° as necessary. Note

that, the saturation 53‘” , Saw or 83° were substituted in the form of Equation 3-2 and
Equation 3-4. As such the relative permeability functions were expressed in term of the
capillary pressure, base parameters and interfacial tension. This expanded function was
used to determined the relative permeability at a given capillary pressure. This function
also served as the base function to propagate the error of the base parameters and the
scaling factors in order to determine the uncertainty limits. Similar procedures were

employed to construct the rest of the constitutive functions and to determine their

uncertainty limits.

81

In general, method 3 and method 1 followed similar procedures to construct the
constitutive curves. The base parameters used in method 3, however, were the air-oil

parameters obtained from the individual regression while those used in method 1 were
from the pooled regression. Each of the saturation functions Saw , S§° or Saw in method
1 has its own base parameter set. Therefore, these curves were constructed in a similar
manner to those used in the case of 53‘” in method 2.

As shown in the previous section, the three-phase oil saturation (SSW) and

relative permeability (kfém) functions are formulated as ﬁmctions of the oil-water and

air-oil capillary pressures (how and hm). When how and ha0 are varied, a family of 53°“

or k3,?“ curves are obtained (Figures 3-5a, 3-5d, 3-6a, 3-6d and 3-7). Among these
curves, two which allow the widest range of oil saturation change were selected for

further analysis. The selected curves were referred to as vadose zone and saturated zone

curves. The vadose zone curves were plotted by increasing hao while holding h0W

constant at high capillary pressure (Figures 3-5b, 3-5e, 3-6b and 3-6e for SS” and

Figures 3-8a, 3-8b, 3-9a, and 3-9b for 1:33“). These plots represent oil desaturation as the
air saturation increased at constant low water saturation. In the ﬁeld, this case might
occur in the vadose zone when the water phase is at residual saturation and the oil phase

percolates as it is driven by gravitational force. The saturated zone curves were plotted

reversing the vadose zone situation, ie., how was increased but h,lo was held constant at

82
low capillary pressure (3-5c, 3-5f, 3-6c and 3-6f for SS” and Figures 3-8c, 3-8d, 3-9c,
and 3-9d for k3,?" ). In this setting the oil saturation was increased as water desaturation
occurs at constant high total liquid saturation. Such a situation may occur when the
DNAPL penetrates below the capillary fringe zone and replaces the water volume in the

soil pore.

Table 3-2: Summary of the agreement between prediction outcomes
by methods 2 or 3 to method 1

 

 

 

 

 

Ottawa Sand : Metea Soil
Method 2 I Method 3 1 Method 2 } Method 3
Saw - i O i - i 0
53° 0 i- ' i- O i- '
SW 0 i- 0 i- O i- O
...‘l .......... -1 ........ 1. ....... 1. ....... 1. ........
kg? 0 | . I O | O
---------------------- ,__-___-,____-__,_____-_.
km 0 1 . l O 1 O
_JSL ___________________ +— _______ 1. _______ +— ........
kg: 0 . o . o . e
“an; ------------------ ‘l" ------- 1" ------- '1‘ --------
Adamant“: ...... 0. ___,'____'___-I--_0____.',___9_--.
S3” 1n sat’d zone 0 1|_ (9 r- 0 1I” @
£on9329532112--____°____,'___.'___-I_-_C3___,'_-__C?.___.
kaow in sat’d zone 0 ' . ' Q ' G)

 

 

Note: 0 poor agreement, no overlapping uncertainty limits in most of the range.
G) fair agreement, overlapping uncertainty limits.
0 good agreement, the mean laid within the uncertainty limit of method 1.
- no prediction required
Table 3-2 summarizes the subjective interpretation of the agreement between
method 1 and 2 as well as between method 1 and 3 as observed on the obtained curves.
The agreement levels in this table were classiﬁed into three categories. A good agreement

score was given for the cases where the mean responses of the prediction curves (method

2 or 3) were laid within the uncertainty limits of the reference curve (method 1). A fair

83

agreement score was given for the cases in which the mean of the prediction curve was
slightly off of the uncertainty limits of the reference curve, however, their uncertainty
limits overlapped each other in all of the observed ranges. If those uncertainty limits were
off at some parts of the observed ranges, a poor agreement score was given.

The Ottawa sand and Metea soil columns in Table 3-2 contrast the effect of soil
texture. The columns labeled as Method 2 and Method 3 show the effects of using
air-water and air-oil as the system basis for prediction, respectively. The arrangement of
the constitutive variables listed in the rows of Table 3-2, from top to bottom, were sorted

according to the order of the procedure followed in constructing the two- and three-phase
k,-S-P relationships. The variables which were potentially affected more by the error in
the base parameters propagated along the prediction were placed in the lower rows. The
subsequent sections discuss the issue associated with soil texture, and are followed by

discussion concerning the methods. The characteristic of the error propagation along the

prediction procedure is discussed in the last sections.

3.4.1. Comparison Across the Soil Types
Table 3-2 shows that, when the Metea soil was used as the porous medium, all of
the obtained k,-S-P curves were in the categories of good or fair. When the Ottawa sand
was used, however, the outcomes range from good to poor. It appears that the soil texture

affects the prediction outcomes. The Metea soil has loamy sand texture while the size of

the Ottawa sand grains are practically uniform. Study by McCuen et al. (1981), based on

84
1,085 samples measured by Rawls et al., (1976) and Holtan et al., (1968), found that the
standard error of the squared root of the Brooks and Corey’s (1964) A. (which is
equivalent to van Genuchten ’s parameter n) for sand was 0.039. Compared to 0.013 for
loamy sand, that for the sand was 300% higher.

The pooled regressions regressed the air-water, oil-water and air-oil
simultaneously. As such, any factor associated with a two-phase type involved was
lumped. Accordingly, any differences in the estimated parameters including their
statistical properties should be related to the soil type. The entries under sub-heading
Method 1 in Table 3-1 list the conﬁdence limits of the P-S parameters estimated by the
pooled regression for the investigated soil. It is observed in the table that the conﬁdence
limits of all parameters obtained in the Ottawa sand were in almost in all cases wider than
those measured in the Metea soil. A wider conﬁdence limit suggests a higher possibility
of obtaining various outcomes when a number of data sets are used to estimate the
parameters. This explaines the pattern exhibited by the scoring outcomes across the soil
type in Table 3-2. In Ottawa sand cases, where the conﬁdence limits of the base
parameters used were relatively wide, the outcomes were varied in a wider range of
outcomes: from good to poor. A favorable outcome may be obtained (Method 2-Ottawa
sand in Table 3-1). Yet, there was also a risk of obtaining unfavorable outcome (Method
3-Ottawa sand in Table 3-1). In contrast, the outcomes of the Metea soil cases, where the
conﬁdence limits of the base parameters used were relatively narrow, were consistently
favorable. This evidence suggests that when prediction via scaling theory was

implemented in a uniform graded porous medium the obtained result may vary. At the

85

extreme case, an unfavorable result such as those observed in the case of Ottawa sand

might be obtained.

3. 4. 2. Comparison Across the Methods

Across the methods, as observed in Table 3-2, method 2 received a greater
number of good scores compared to method 3. This tendency was consistently observed
in both soils. This scoring outcome might suggest that method 2 is a better substitute for
method 1 than is method 3. The major difference between method 2 and 3 was in the two-
phase type from which the base parameters were determined. If it is true that the
superiority of method 2 over method 3 is due to the use of the two-phase type, then a
parallel tendency should also be observed in their base parameters.

The following discussion inspects any possible correlation between the base

parameters used in methods 2 and 3 and the superiority of method 2 over method 3.

Except the scaling factor B, the other parameters S,, n and Claw (method 2) or Ciao

(method 3) were inspected. At this stage, each of these parameters is assumed to have

potential for inﬂuencing the difference in the scores across the two methods. As shown in

Equation 3-5 and Equation 3-6 the scaling factors B were involved in the scaling
procedure. These factors, however, were determined using the interfacial tension data and

the same interfacial tension data was used by the two methods. Based on this, it is

unlikely that B has potential to inﬂuence the difference between methods.

86

Table 3-3: Discrepancies between the base parameters used in
method 2 and 3 relative to those used in method 1.

 

 

 

Base Ottawa sand Metea soil
Parameters Method 2 Method 3 Method 2 Method 3
S, 27 % 20 % 5 % 2 %
a“, 1 % - 4 % -
a“, — 26 % - 8 %
n 5 % 31 % 9 % 7 %

 

Table 3-3 shows the discrepancies of the base parameters S,, n and (law or 0:30

employed by the two methods relative to the base parameters used in method 1. The

magnitude of these descrepancies are regarded as the level of the error of those
parameters. As observed in this table, the discrepancies in the parameters S, and n were
not consistent with the scoring outcome observed in Table 3-2. For example, while

method 2 received a better score than method 3, the errors of the S, employed in method

2 was higher than those employed in method 3.

That S, and n cannot be associated to the two-phase type involved appears to be in

agreement with the assumption which states that these parameters are porous medium
dependent parameters rather than the two-phase type (Lenhard and Parker, 1987). Note

that, methods 2 and 3 estimate the P-S parameters using the individual regression

procedure. This procedure did not pool the data according to the soil type. The S, and n
used in method 2 were obtained from air-water measurements while those used in method

3 were from air-oil measurements. To permit more general inspection, S, and n estimated

by the individual regression upon each of the air-water, air-oil and oil-water data were

87

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l 1- n-
r H
... 1
i .... l l'
J 1
l7 ... l L
r7 5 "
. 1
i ﬁ u r
c i *-
I I
\O V ("I N
ll
1 n l ..
I u 7
1 2 ~
—u p
l n I _
l H l
I I I I
In ”I "1 N. "I Q
d c o c c o

 

Metea Soil
Air-Oil
Metea Soil
Oil-Water

Metea Soil
Air-Water

Metea Soil
Air-Oil

Metea Soil
Oil-Water

Metea Soil
Air-Water

Ottawa Sand
Air-Oil

Ottawa Sand
Oil-Water

Ottawa Sand
Air-Water

Ottawa Sand
Air-Oil

Ottawa Sand
Oil-Water

Ottawa Sand
Air-Water

Figure 3-10: Parameters S, and n estimated by the individual regression with their 95% conﬁdence limits

I-rtedr

88
plotted in Figure 3-10. With the individual regression procedure, neither their

independence to the two-phase type nor their correlation to the soil type was artiﬁcially

imposed. However, as observed in Figure 3-10, except in the case of "are of the Ottawa
sand, S, and n appear to be more consistent across the soil types than across the methods.

Based on all of the mentioned evidence, it is concluded that S, and n are independent of

the two-phase type used. As such, the outcome across the two methods observed in Table

3-2 was unlikely to be the signature of the errors of S, and n. Unfavorable scores in the

cases of method 2 using Ottawa sand migh be strongly inﬂuenced by the error in nao. ‘

However, it is viewed as more characteristic of the soil type rather than the methods.

The discrepancies in the parameter at in Table 3-3, on the other hand, appear to be
correlated to the differences in the scores obtained by the two methods. Method 2, which
received overall good results had or discrepancies of only 1% and 4%. Method 3,
however, received overall poorer results and had at discrepancies of 26% and 8%. It
appears that good result might be correlated to low discrepancies, whereas poorer results
to high discrepancies.

As concluded in the previous section, S, and n appeared to be porous medium
dependent parameters. The pooled regression accommodates this condition. Thus, oraw

cow and am obtained via pooled regression are regarded as the most representative on. As

such, any discrepancies relative to these on can be used to measure the level of errors

89

embodied in or. Additionally, the width of the conﬁdence limits of the or obtained by the
pooled regression indicates the range in which different outcomes from different data sets
might be obtained. The wider these limits the more the risk of obtaining higher

discrepancies. In turn it also indicates higher risk of obtaining stronger errors.

The parameters listed under sub-heading Method 1 in Table 3-1 were obtained via

pooled regression. An inspection on a values in this list, shows that the order of the

magnitude of a was aao>aow>aaw and the width of the conﬁdence limits of those

obtained by the pooled regression followed the same order. These patterns were

consistently observed in both soils. The observed pattern suggest that the higher the

magnitude of a the lower its precision. A study by van Genuchten and Nielsen (1985)
postulated that or is inversely proportional to displacement capillary pressure(ade '1).
Utilizing this relationship, the observed pattern indicated that low Pd value possesses low

accuracy. Lenhard and Parker (1987) observed similar problems. Their P-S measurement
using benzyl alcohol as the oil phase covering the lowest range of capillary pressures

among the measured two-phase systems, suffered the highest uncertainties.

In the investigated cases, the aaw in method 2 was equivalent to Pd= 43.5 mbars

for the Metea soil and 17.1 for the Ottawa sand. In method 3, 030 was equivalent to 18.7

(Metea soil) and 7.5 mbars (Ottawa sand). In order to anticipate 200 mbar maximum
capillary pressure measurement in the air-oil system of the Ottawa sand, a 5 psi (=345

mbar) rated pressure transducer membrane was used. While it gave a good resolution for

90

low pressure measurement, it lacked accuracy due to a zero shift problem. On the
average, the calibration in the air-oil measurement setting indicated a $0.7 mbar standard

error of estimate. When this standard error of estimate is expressed as 95% conﬁdence

limits it is equivalent to approximately 20% of the Pd of the air-oil system in the case of

Ottawa sand. For the air-water system, on the other hand, it equivalent to only 3.5% of

Pd. Based on this inspection, it is concluded that am was less reliable compared to aaw.
Method 3 used aao while method 2 used oraw. When identical S, and n were

employed in the two methods and since (1a,, was less reliable than oraw, the chance of

obtaining bias in the prediction outcome is higher in method 3 than in method 2. As
observed in Table 3-2, method 2 performed reasonably well not only when a well-graded
soil such as Metea soil was used but also when the poorly graded Ottawa sand was used.
Method 3 appears to be acceptable only when Metea soil was used.

The source of discrepancies between methods 1 and 2 or methods 1 and 3 was due
to the discrepancies in the P-S parameters used. When method 3 was applied using

Ottawa sand, the base parameters used suffered the combination of two problems; less

precision in the parameters S, and n due to the texture of the soil and low accuracy in aao

due to difﬁculty in measuring low capillary pressure. As a result, the parameters
determined solely from the air-oil data measured in a uniform grain soil might not be
representative enough to be used as the scaling reference compared to those derived from

the air-water data.

91

As presented in the previous section the two- and three-phase oil saturation S,, and
relative permeability k,0 are functions of the two-phase oil saturation in an air-oil system
(S,,) and water saturation in an oil-water system (Sow). In method 2 both Sa0 and SW are
prediction curves while in method 3 only S0,, is a prediction curve. Thus, in obtaining So
and k,,,, method 2 would propagate any error embodied in the base parameters more than
would method 3. Based on this, prediction via method 3 should not be inferior when
compared to the prediction via method 2. The scoring outcome on Table 3-2, however,

shows that method 2 was better than method 3. As concluded previously, method 3 has a

problem in determining the parameter aw. Apparently, the difﬁculty of measuring low

capillary pressure has a stronger effect upon the outcome than when this additional source

of error is induced.

3. 4.3. Eﬂ'ect of Propagated Error on the Constitutive Variables

Parker et al. (1987) postulated that prediction of the relative permeability
function might be sensitive to any disturbance in the base P-S parameters. The
arrangement of the rows in Table 3-2, as mentioned previously, roughly reﬂects the
sequential steps in constructing the two- and three-phase constitutive relationship curves.
In turn, it also reﬂects the way the errors in the base parameters and scaling factors were
propagated. The constitutive variables in the lower rows were functions of the variables
in the upper rows. Therefore, the lower row variables possessed the uncertainty of the

variables in the upper rows. Thus, the lower row variables possessed greater uncertainties

92

as the error that embodied in the base parameters and the scaling factors were propagated
further.

The columns under the Metea Soil heading in Table 3-2 show the above trend.
The upper rows, up to variable k3: , were scored in the category good. Below this row,

some scores of fair were observed. This pattern indicates that the errors propagated up to

variable hf: produced only insigniﬁcant bias. Further propagation slightly increased the
bias. The bias resulting from this increment at the end point (the three-phase oil relative
permeability), however, is still within tolerable limits. This was possible as the error
embodied in the original bias was relatively small. Therefore, it follows that the
magnitude of error of the base parameters as tabulated in the pertaining columns in Table

3-3 can be regarded as small.

It is interesting to observe that 27% error in the S,, as shown in the column Ottawa

Sand - Method 2 in Table 3-3, yields insigniﬁcant bias in all of the prediction curves.
This is only possible, however, if the accuracy of the other parameter was high.

Nevertheless, this particular case suggests the insensitivity of the employed prediction

methods employed to the parameter S,. It is speculated that the bias in the prediction

curve associated with error in this parameter is localized in the vicinity of the low
saturation region. On the other hand, even with parameter estimation based on measured
data the uncertainty in this region is high due to the non-linearity of the constitutive
relation ﬁrnction.

The errors of the base parameters in the case of Ottawa sand - Method 3 (Table

3-2) were high. The predictions employing these parameters, as shown in Table 3-2, were

93

unfavorable. Despite the signiﬁcant discrepancy observed in the prediction curves the
errors of the base parameters in this particular case was not clearly evident in the P-S
curves plot. Figure 3-11 was plotted to demonstrate this issue. This ﬁgure plots the
measured data and the ﬁtted P-S curves on these data via the pooled regression (dashed
line) superimposed by the curves obtained via the individual regressions (solid line). The
parameters used by the pooled regression curves were the parameters employed in
method 1. Those used by the individual regression curves were the base parameters used
in methods 2 or 3. Thus, the parameter discrepancies listed in Table 3-3 were the
discrepancies of the parameters used by the dashed and shaded areas.

In Figure 3-11 the discrepancy between the solid and dashed curves in the case of
the air-oil system in Ottawa sand appears to be comparable, for example, to the case of

the air-water system in Metea soil. As a P-S plot, error as high as 20%, 26% and 31% in

parameters S,, n and a appears similar to errors of 5%, 4% and 9% (Table 3-3). If these

errors were included in the scaling procedure, however, a signiﬁcant difference in
outcomes would be clearly observed. In the same example, the air-oil and air-water
systems were scaled to predict the oil-water P—S curve. The resulting curves were the
plots in Figure 3-2a and Figure 3-lc, respectively. It is observed that the plot in Figure
3-2a was signiﬁcantly worse than that in Figure 3-lc. Note that, the errors of the scaling
factors involved in constructing Figure 3-2a was only 0.02% while in Figure 3-1c it was
9.5%. Therefore, it is believed that the errors observed in Figure 3-2a and Figure 3-1c

were strongly affected by the errors in their base parameters rather than to the errors in

their scaling factors.

94

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In the actual implementation of the scaling theory, only one system is required to
be measured. When data is available from only one system, the goodness of the obtained
base parameters can only be veriﬁed through an inspection of the agreement upon the
best estimated P-S curve against the data points. As demonstrated above, this technique

might not be sufﬁcient to reveal even errors as high as 20%, 26% and 31% in parameters

S,, n and on. Based on this, in order to increase the possibility of obtaining more

representative parameters, it is necessary to replicate the sample.

3. 5. SUMMAR YAND CONCLUSION

Parker et al. (1987) proposed a predictive method employing scaling theory.
Using their theory, a complete set of two- and three-phase k,-S-P relationships can be
constructed based only on one set of P-S parameters, referred to as the base parameters,
measured in one two-phase system plus the pertaining interfacial tension data. Two

alternatives to implement this theory were investigated, referred to as method 2 and 3

respectively. Method 2 uses the air-water system to determined the base parameters while

Method 3 uses the air-oil system.

Method 2 was regarded as the most efﬁcient alternative in the case where

air-water P-S data are available. The base parameters S,, n, (law were determined from the
available data and the parameters (low and cm were predicted via interfacial tension

measurements. Method 3 was similar to method 2, but the S,, n, Ciao were determined

96

from air-oil P-S data while the (raw and aow were predicted. This alternative might be

suitable in cases where sufficient data are not available, yet more efficient procedures

than those used in method 2 are desired.

These methods were tested to predict the two- and three phase k,-S-P relationships

using two contrasting soils: uniform grain Ottawa sand and loamy sand Metea. As the
reference to justify the performance of methods 2 and 3, the obtained prediction curves

were compared to those obtained via method 1 which is a predictive method without
employing the scaling theory.

When the Metea soil was used as the porous medium, the prediction of the k,-S-P

curves yielded a favorable outcome. For the cases where Ottawa sand was used the
outcomes vary. At the extreme case, an unfavorable result such as those observed in the
case of Ottawa sand with method 3 might be obtained. Inspection of the statistical
properties of the base parameters, suggests that determination of these parameters in a
poorly graded soil, such as Ottawa sand, might result in a high variation. In turn, the risk

of obtaining less representative parameters is also high.

The investigated cases suggest that the parameters S, and n were independent of

the two-phase system used. The deference between the outcomes obtained via methods 2
and 3 was associated with the errors in their values of the parameter 0:. Determination of

or in the air-water system was more reliable than in the air-oil system. This was due to the

97

difﬁculty of measuring low capillary pressure. As a result, prediction by method 2 which
is based on air-water data appears to be superior over method 3.

When method 3 was applied using Ottawa sand, the base parameters used suffered

the combination of two problems; less precision in parameters S, and n due to the texture

of the soil and low accuracy in am due to the difficulty in measuring low capillary

pressure. As a result, the parameters determined solely from the air-oil data measured in a
uniform grain soil might not be representative enough to be used as the scaling reference
compared to those derived from the air-water data.

A less representative parameters set may not deteriorate in agreement between the
data points and the regressed P-S curve, however, it may cause a problem if used as the
basis of prediction. Hence, the use of method 3 should be limited to a well-graded soil.
For a uniform grain type soil, it is more rigorous to employ method 2 which gives better
assurances in providing representative parameters. Furthermore, the issue of difﬁculty in
measuring low capillary pressure inherent in method 3 has a more signiﬁcant effect than

the error propagated by the additional step required in method 2.

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Beck, J .V., and KJ. Arnold. 1976. Parameter Estimation in Engineering and Science.
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Bethel, FT, and J .C. Calhoun. 1953. Capilary desaturation in unconsolidated beads.
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Corey, A.T., and RH. Brooks. 1975. Drainage characteristics of soils. Soil Sci. Soc.
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ﬂow: 11. Flow system data. Soil Science, 23 :434-437

APPENDICES

A. PRESSURE-SATURATION MEASUREMENT
SINIULATION

A. 1. Model Conceptualization.
A.2. Simulation Flowchart
A.3. Program Listing in Excel Worksheet.

102

103

A1 MODEL CONCEPTUALIZATION

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FLOWCHART
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SIMULATION SCENARIO
Initial Condition

Continue current state
to
with
Pw z=0 :

cm
suction:

cm
Redistribution Criteria
Switch to redistribution at

time:
Outflow

second

cm3

0K

Canoe!

Reset Tune/omFlow

 

138

B. PRESSURE-SATURATION DATA OBTAINED USING
SATURATION EQUILIBRIUM METHOD

B. l. Air-Water Data
3.2. Air-Oil Data

139

 

 

 

 

 

 

 

PROJECT ME—109a

SOIL TYPE Metea <850u

SYSTEM Air 0“

PROC. NOTE Reff. for air-oil

DATE 09-Jul-90

rho 1.622 @ 20 deg

PREPARATION DATA 1

W by ﬁlter Paper

SAMPLE WEIGHT OF Oil
NO Air Dry Saturated absorbed
1 0.162 0.6832 0.5212
2 0.184 0.7808 0.5968
0.5g
PREPARATION DATA 2
Initial Moisture Content
SAMPLE Tin Air dry Oven dry lnit.
NO + Tin + Tin Moisture
1 1.379 5.304 5.250 0.014
2 1.387 3.551 3.510 0.019
3 1.378 2.398 2.382 0.016
0.016399478

   

 

PREPARATION DATA 3 (in gram)
CAPS: 3.994 ram
CELL RING ALL SATURATED 8 AT THE
LABEL CELL CELL,CAPS END OF
PART BOTTOM FILLED EXPERIMENT
<1) <2> <3> <4>
1 70.43 409.45 582.44 549.23
2 70.31 406.71 582.05 #NIA
3 70.59 407.70 582.95 549.71
4 71.96 407.59 583.03 550.16
5 70.54 406.52 583.13 550.95

M
NOTE: Col<1>: annm+wmu+ Rimg+Upp¢part+ Lwrpan+0vn¢yplu+2ADFiham
Col<3>z<2>+batomcqu+amuodooil+wbellawﬁllod

FINAL PROC DATA
For Residual Moisture Calculation
CELL TIN TIN, BRAS TIN, BRAS
LABEL RING, MO- RING, OVN
IST SMPL DRY SMPL
<5> <6> <7>
1 6.97 186.89 176.37
2 7.71 #N/A #NIA
3 6.97 187.03 176.45
4 6.82 188.61 177.84
5 6.9L 187.84 176.64

 

140

 

 

 

 

 

 

PRESSURE DATA
Pressure ' 7.5 7.5 15 15 30 30
Date 12-Jul 13-Jul 13-Jul 14-Jul 14—Jul 16-Jul
Time 01:14 PM 11:28 AM 11:29 AM 12:25 PM 02:47 PM 10:20 AM
Temperature : 22 23 23 23 23 _23
Step: —START- --END—- -START- -—END— -START- —-END—-
CELL <8a> <8a> <Bb> <8b> <8c> <8c>
1 9.74 11.87 11.87 21.01 9.47 21.40
2 9.55 11.14 11.14 19.61 9.48 19.65
3 10.65 15.60 15.60 24.12 9.18 17.70
4 10.43 12.11 12.11 20.96 9.30 19.13
5 9.12 11.10 11.10 19.21 10.31 20.72
Pressure 40 40 50 50 75 75
Date 16-Jul 17-JuI 17-Jul 18-Jul 18-Jul 19-Jul
Time 12:16 PM 10:20 AM 10:27 AM 11:50 AM 11:50 AM 09:16 AM
Temperature : 23 23 23 23 23 23
Step: ~START— --END-— —START- -END-—- -START— -END—
CELL <8d> <8d> <8a> <8e> <8f> <8f>
1 9.54 12.45 12.45 16.76 16.76 17.62
2 9.34 #N/A #N/A #N/A #N/A #N/A
3 9.66 13.16 13.16 15.82 15.82 17.65
4 9.07 13.88 13.88 17.14 17.14 18.09
5 10.50 16.31 16.31 18.49 18.49 19.14
Pressure 100 100 238 238
Date 19-Jul 20-Jul 20-Jul 21-Jul
Time 10:39 AM 10:33 AM 10:39 AM 10:33 AM
Temperature : 23 23 23 23
Step: -START- -END—- -START- —END—-
CELL <8g> (89> <8h> <8b>
1 9.62 11.52 1 1.52 12.29
2 #NIA #NIA #NIA WA
3 9.65 12.17 12.17 12.41
4 9.09 10.67 10.67 11.97
5 10.14 11.85 11.85 12.82
DATA REDUCTION 1
Bulk Densil
OVEN DRY BULK
CELL SOIL WGT DENSITY
<10>
1 1.453
2 WA
3 1.452
4 1.454
5 1.456
Note: Ring Vol: 68.119 cm3

 

 

141

 

 

 

 

 

DATA REDUCTION 2
ResidualMorsture Content _
RESIDUAL INITIAL WATER WATER MASS
CELL 140151qu MOISTURE RELEASED COLLECTED BAIJANCE
comm comm
<11>=6-7 <12>=11+13 <13>=34 <l4>=sum(d8) «9:04.133:
1 10.52 43.73 33.21 33.95 1.69
2 #NIA #NIA #N/A #N/A #N/A
3 10.58 43.82 3324 32.74 -1.14
4 10.77 43.64 32.87 32.26 -1.40
5 11.20 43.38 32.18 31.82 -0.83
WEIGHT OF MOISTURE LOSS
PRE: 7.5 15 30 40 50 75 100 238
CELL
AT EACH STEP
1 2.13 9.14 11.93 2.91 4.31 0.86 1.90 0.77
2 1.59 8.47 10.17 #NIA #NIA WA #NIA #NIA
3 4.95 8.52 8.52 3.50 2.66 1.83 2.52 0.24
4 1.68 8.85 9.83 4.81 3.26 0.95 1.58 1.30
5 1.98 8.11 10.41 5.81 2.18 0.65 1.71 0.97
ACCUMULATIVE
1 2.13 9.14 11.93 2.91 4.31 0.86 1.90 0.77
2 3.72 17.61 22.10 #NIA #NIA #NIA #NIA WA
3 7.08 17.66 20.45 6.41 6.97 2.69 4.42 1.01
4 3.81 17.99 21.76 7.72 7.57 1.81 3.48 2.07
5 4.11 17.25 22.34 8.72 6.49 1.51 3.61 1.74

 

 

 

 

1 0.402 0.383 0.300
2 #N/A #NIA WA
3 0.392 0.347 0.270
4 0.389 0.374 0.294
5 0.389 0.371 0.298
AVG #NIA #NIA #NIA
STD‘ NA NA #NIA
DEGREE OF SATURATION
1 1.00 0.95 0.75
2 NA NA #NIA
3 1.00 0.89 0.89
4 1.00 0.98 0.78
5 1.00 0.95 0.77

 

0.193
#NIA
0.193
0.205
0.204
WA

#NIA

0.188
NA
0.181
0.182
0.151
#NIA

WA

0.41
#N/A
0.41

0.42
_ 0.39_

0.127
#N/A
0.137
0.132
0.132
#N/A
NA

.32
#N/A
0.35
0.34

0.119
#NIA
0.121
0.124
0.128
NA
NA

0.30
#NIA
0.31

0.32

0.25
#NIA
0.25
0.28

0.095
#NIA

0.098
0.097
0.101
#NIA

#NIA

0.24
WA
0.24
0.25

142

 

PROJECT ME-109
SOIL TYPE Metea
SYSTEM Air Water
PROC. NOTE Replica #1 of 108 reff. for aw
DATE 29-Sep-90
rho 1.021 @ 20 deg
PREPARATION DATA 1
Wei ht of water absorbed b ﬁlter r
SAMPLE WEIGHT OF Water
NO Air Dry Saturated absorbed
1 0.171 0.427 0.256
2 0.194 0.488 0.294

 

 

     

 

 

SAMPLE Tin Moist Oven dry lnit.
NO soiI+Tin + Tin Moisture
1 0.920 7.210 6.860 0.059
2 0.920 3.280 3.150 0.058
3 0.910 4.190 4.040 0.048
0.055047282
PREPARATION DATA 3 (in gram)
CAPS: 8.201 Tram

   

__ “ALL _“ATTRATE' "AT T~ ’
CELL. RING CELL CELL,CAPS END OF

 

 

LABEL PART DTTOM FILLE EXPERIMENT
<1> <2> <3> <4>

1 70.39 399.76 552.99 532.29

2 73.27 398.88 555.03 532.23

3 69.43 397.74 555.70 534.13

4 72.81 397.08 553.85 533.78

5 73.50 393.05 552.38 531 .33

M
NOTEICOI <1>: Wingnuts+Washa+ijg+Uppaput+Lwrpu1+0vn¢yplatc+2ADFﬂwpaper
Col<3>:<2>+bottomcaps+aaunatcdsoi1+apaocbeﬂow

FINAL PROC DATA
For Residual Moisture Calculation

w
CELL . TIN TIN, BRAS TIN, BRAS

 

 

LABEL RING, MO- RING, OVN
IST SMPL DRY SMPL
<5> <6> <7>
1 3.83 177.51 172.51
2 3.83 180.44 174.43
3 3.83 176.40 171.74
4 3.81 180.52 175.99
5 3.81 181.36 176.19

   

143

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

DATA REDUCTION 2
_R_e§°dual Moisture Content _ _
_ . RESIDUAL INITIAL ¥ WATER 7 WATER MASS
CELL MOISTURE MOISTURE RELEASED COLLECTED BALLANCE
CONTENT CONTENT
<11>=6-7 <l2>=l1+l3 <13>=34 <l4>=sum(d8) <15>=(I4—13)/12
1 5.00 25.70 20.70 21.13 1.88
2 8.01 28.81 22.80 22.53 -0.94
3 4.88 28.23 21.57 21.12 -1.72
4 4.53 24.80 20.07 20.19 0.49
5 5.17 28.22 21.05 21.70 2.48
WEIGHT OF MOISTURE LOSS
PRESS 17 . 49 88 89 117 187 317 717
CELL
AT EACH STEP
1 0.73 5.80 8.97 1.97 1.42 2.07 2.04 0.33
2 0.92 1.72 8.80 5.03 2.74 1.18 2.08 0.08
3 0.80 8.40 8.40 1 .53 2.89 0.93 1 .80 0.77
4 0.98 5.99 8.84 1.43 2.01 1.13 1.74 0.29
5 1.09 1.02 9.82 4.03 1.72 0.95 2.58 0.71
ACCUMULATIVE
1 0.73 8.33 13.30 15.27 18.89 18.78 20.80 21.13
2 0.92 2.84 11.44 18.47 19.21 20.39 22.45 22.53
3 0.80 7.00 13.40 14.93 17.82 18.55 20.35 21.12
4 0.98 8.95 13.59 15.02 17.03 18.18 19.90 20.19
5 1.09 2.11 11.73 15.78 17.48 18.43 20.99 21.70
VOLUMETRIC MOISTURE CONTENT
PRESS 0 17 49 88 89 117 187 317 717
CELL
1 0.384 0.373 0.291 0.188 0.159 0.139 0.108 0.078 0.073
2 0.419 0.408 0.380 0.251 0.177 0.137 0.120 0.089 0.088
3 0.378 0.370 0.278 0.182 0.159 0.120 0.108 0.080 0.088
4 0.383 0.349 0.281 0.183 0.142 0.113 0.098 0.071 0.087
5 0.394 0.378 0.383 0.222 0.183 0.138 0.124 0.088 0.078
AVG 0.388 0.375 0.314 0.201 0.180 0.129 0.111 0.081 0.075
STDV 0.019 0.018 0.048 0.031 0.011 0.011 0.010 0.007 0.008
DEGREE OF SATURATION
1 1.00 0.97 0.78 0.49 0.42 0.38 0.28 0.20 0.19
2 1.00 0.97 0.91 0.80 0.42 0.33 0.29 0.21 0.21
3 1.00 0.98 0.73 0.48 0.42 0.32 0.28 0.21 0.18
4 1.00 0.98 0.72 0.45 0.39 0.31 0.27 0.20 0.18
5 1.00 0.98 0.92 0.58 0.41 0.35 0.31 0.22 0.19

 

144

 

PRESSURE DATA

Pressure : 1 7 49 66

Date : 01-Aug 02-Aug 02-Aug 03-Aug 03-Aug 05-Aug
Time : 01:40 PM 11:28AM 11:28AM 11:17AM 11:17AM 01:30AM
Temperature : 22 23 23 23 23 23
Step: «START— --END-— -START- --END-- -START-— --END-—-
CELL

 

10.40 11.13 11.13 18.73 9.35 18.32
9.37 10.29 10.29 12.01 9.88 18.68
9.65 10.25 10.25 16.85 9.45 15.85

10.01 10.97 10.97 18.98 10.12 18.76
9.32 10.41 10.41 11.43 9.18 18.80

I

IUI&00Nd

 

 

T>ressure : 89 117'? 167
Date : 05-Aug 06-Aug 06-Aug 07-Aug 07-Aug 08-Aug
Time : 01:30 AM 11:34 AM 11:34 AM 01:57 PM 01 :57 PM 01:57 PM
Temperature : 23 23 23 23 23 23
Step: -START- --END- -START- --END— -START—- --END—
CELL
1 9.65 11.62 11.62 13.04 13.04 15.11
2 9.12 14.15 14.15 16.89 16.89 18.07
3 9.56 11.09 11.09 13.78 13.78 14.71
4 9.88 11.31 11.31 13.32 13.32 14.45
5 9.21 13.24 13.24 14.96 14.96 15.91

   

 

 

_Pressure : 317 717

 

     

 

 

Date : 08-Aug 09-Aug 09-Aug 10—Aug
Time : 01:57 PM 12:50 PM 12:50 PM 12:50 PM
Temperature : 23 23 23 23
Step: -START- --END-—- -START- --END—-
CELL

1 15.11 17.15 9.43 9.76

2 18.07 20.13 9.45 9.53

3 14.71 16.51 10.05 10.82

4 14.45 16.19 10.11 10.40

5 15.91 18.47 9.32 10.03
DATA REDUCTION 1
Bulk Densit

OVEN DR‘ BULK
CELL SOIL WG' DENSITY
<9> <10>

1 98.29 1.443

2 97.33 1.429

3 98.48 1.446

4 99.37 1.459

5 98.88 1.452

m
Note: Ring Vol: 68.119

145

C. REGRESSION OUTPUT

C. l. Air-Water System, Based on Data Measured Using Saturation Equilibrium
C.2. 145133;.“ System, Based on Data Measured Using Pressure Equilibrium
C.3. lAiiret-IOIIOdSystem, Based on Data Measured Using Saturation Equilibrium
C.4. $31an System, Based on Data Measured Using Pressure Equilibrium

146

SPSS Result
Air-Hater System

A” Saturation Equilibrium.

Ami. Cell 1
Nonlinear Regression Summary Statistics Dependent Variable SE_Sl
Source DF Sum of Squares Mean Square
Regression 3 3.21827 1.07276
Residual 6 2.095629E-03 3.492714E-04
Uncorrected Total 9 3.22037
(Corrected Total) 8 .79303
R squared = 1 - Residual SS / Corrected SS = .99736
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .185867309 .016738882 .144908741 .226825877
A .023725148 .001067714 .021112546 .026337751
N 2.608953612 .153587725 2.233137989 2.984769236
Asymptotic Correlation Matrix of the Parameter Estimates
SR A. N
SR 1.0000 -.1204 .7100
A -.1204 1.0000 -.6550
N .7100 -.6550 1.0000
1.2. Cell 2
Nonlinear Regression Summary Statistics Dependent Variable SE_SZ
Source DF Sum of Squares Mean Square
Regression 3 3.59751 1.19917
Residual 6 1.829279E-03 3.048798E-04
Uncorrected Total 9 3.59934
(Corrected Total) 8 .85753
R squared = 1 - Residual SS / Corrected SS = .99787
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .223956252 .012046083 .194480548 .253431956
A. .017169866 .000465797 .016030102 .018309629
N 3.759266303 .232879374 3.189431003 4.329101604
.Asymptotic Correlation Matrix of the Parameter Estimates
SR A. N
SR 1.0000 .0872 .5541
.A .0872 1.0000 -.5472
N .5541 -.5472 1.0000
Am3. Cell 3
Nonlinear Regression Summary Statistics Dependent variable SE_53
Source DF Sum of Squares Mean Square
Regression 3 3.13448 1.04483
Residual 6 2.192198E-03 3.653664E-04
Uncorrected Total 9 3.13667
(Corrected Total) 8 .80941

R squared = 1 - Residual SS / Corrected SS = .99729

147

Asymptotic 95 8

Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .179457012 .016845170 .138238366 .220675657
A .024686491 .001141628 .021893028 .027479954
N 2.611190050 .156043455 2.229365469 2.993014630
Asymptotic Correlation Matrix of the Parameter Estimates
SR A N
SR 1.0000 -.1260 .7096
A -.1260 1.0000 -.6586
N .7096 -.6586 1.0000
A.4. Cell 4
Nonlinear Regression Summary Statistics Dependent variable SE_S4
Source DF Sum of Squares Mean Square
Regression 3 3.01430 1.00477
Residual 6 1.613228E-O3 2.688714E-04
Uncorrected Total 9 3.01592
(Corrected Total) 8 .82072
R squared = 1 - Residual SS / Corrected SS = .99803
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .175272285 .013755495 .141613801 .208930769
A .025508022 .000986397 .023094396 .027921649
N 2.681842856 .137600831 2.345145753 3.018539960
Asymptotic Correlation Matrix of the Parameter Estimates
SR A. N
SR 1.0000 -.1170 .6982
A. -.1170 1.0000 -.6552
N .6982 -.6552 1.0000
.A.5. Cell 5
Nonlinear Regression Summary Statistics Dependent variable SE_SS
Source DF Sum of Squares Mean Square
Regression 3 3.53873 1.17958
Residual 6 7.839523E-03 1.306587E-03
Uncorrected Total 9 3.54657
(Corrected Total) 8 .86994
R squared = 1 - Residual 38 / Corrected SS = .99099
Asymptotic 95 %
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .219392728 .025020222 .158170451 .280615005
A. .017738378 .001020385 .015241585 .020235171
N 3.674507015 .461516478 2.545216874 4.803797155
Asymptotic Correlation Matrix of the Parameter Estimates
SR .A N
SR 1.0000 .0830 .5612

.A .0830 1.0000 -.5479
N .5612 -.5479 1.0000

148

ASS. 83 all
Nonlinear Regression Summary Statistics Dependent Variable SE_S
Source DF Sum of Squares Mean Square
Regression 3 16.45007 5.48336
Residual 42 .06880 1.637981E-03
Uncorrected Total 45 16.51887
(Corrected Total) 44 4.17228
R squared = 1 - Residual SS / Corrected SS = .98351
Asymptotic 95 5
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .193063491 .014628308 .163542370 .222584612
A .021273045 .000791200 .019676340 .022869751
N 2.907024543 .168339440 2.567301800 3.246747286
Asymptotic Correlation Matrix of the Parameter Estimates
SR A N
SR 1.0000 -.0372 .6603
A. -.0372 1.0000 -.6124
N .6603 -.6124 1.0000

8. Preaure‘lquilibriumi

8.1. Cell 1
Nonlinear Regression Summary Statistics Dependent Variable PE_Sl
Source DF Sum of Squares Mean Square
Regression 3 8.72360 2.90787
Residual 15 .01589 1.059621E-03
Uncorrected Total 18 8.73949
(Corrected Total) 17 1.17524
R squared = 1 - Residual SS / Corrected 55 = .98648
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .265765467 .022804424 .217158987 .314371947
A. .021889159 .000655873 .020491200 .023287119
N 3.134065543 .258524024 2.583034630 3.685096456
Asymptotic Correlation Matrix of the Parameter Estimates
SR A. N
SR 1.0000 .3154 .7042
.A .3154 1.0000 -.1477
N .7042 -.1477 1.0000
3.2. Cell 2
Nonlinear Regression Summary Statistics Dependent Variable PE_82
Source DF Sum of Squares Mean Square
Regression 3 8.18419 2.72806
Residual 15 .01149 7.657700E-04
Uncorrected Total 18 8.19568
(Corrected Total) 17 1.15263

R squared = 1 - Residual SS / Corrected SS = .99003

149

Asymptotic 95 %

Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .265465274 .016785044 .229688800 .301241747
A .024563224 .000477704 .023545022 .025581426
N 3.943358281 .271261578 3.365177913 4.521538649
Asymptotic Correlation Matrix of the Parameter Estimates
SR A N
SR 1.0000 .3820 .6094
A .3820 1.0000 -.1490
N .6094 -.1490 1.0000
8.3. Cell 3
Nonlinear Regression Summary Statistics Dependent Variable PE_S3
Source DE Sum of Squares Mean Square
Regression 3 8.46954 2.82318
Residual 15 .01051 7.008422E-04
Uncorrected Total 18 8.48006
(Corrected Total) 17 .94411
R squared = 1 - Residual SS / Corrected SS = .98887
Asymptotic 95 %
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .236929406 .025652718 .182251932 .291606881
A .023017836 .000700355 .021525065 .024510607
N 2.724943257 .165280030 2.372657212 3.077229303
Asymptotic Correlation Matrix of the Parameter Estimates
SR A. N
SR 1.0000 .3217 .7502
A. .3217 1.0000 -.2121
N ' .7502 -.2121 1.0000
8.4. Cell 4
Nonlinear Regression Summary Statistics Dependent variable PE_S4
Source DF Sum of Squares Mean Square
Regression 3 8.76843 2.92281
Residual 15 .02138 1.425210E-03
Uncorrected Total 18 8.78981
(Corrected Total) 17 .98161
R squared = 1 - Residual 55 / Corrected 33 = .97822
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .224802526 .025355605 .170758333 .278846720
A. .020010398 .000487883 .018970500 .021050296
N 3.908730532 .361630200 3.137934007 4.679527058
Asymptotic Correlation Matrix of the Parameter Estimates
SR .A N
SR 1.0000 .4754 .4657
A .4754 1.0000 -.2153
N .4657 -.2153 1.0000

150

 

8.5. Cell 5
Nonlinear Regression Summary Statistics Dependent Variable PE_SS
Source DE Sum of Squares Mean Square
Regression 3 8.37428 2.79143
Residual 15 .01095 7.299417E-04
Uncorrected Total 18 8.38523
(Corrected Total) 17 1.00163
R squared = 1 - Residual SS / Corrected SS = .98907
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .213951001 .023159461 .164587779 .263314223
A .019195945 .000491686 .018147941 .020243949
N 2.996441761 .184490669 2.603209209 3.389674313 ’
Asymptotic Correlation Matrix of the Parameter Estimates
SR A N i
SR 1.0000 .3411 .6886 I
A .3411 1.0000 -.2401 L
N .6888 -.2401 1.0000
8.6. Cell 6
Nonlinear Regression Summary Statistics Dependent variable PE_S6
Source DF Sum of Squares Mean Square
Regression 3 8.31781 2.77260
Residual 15 .03086 2.0576088-03
Uncorrected Total 18 8.34867
(Corrected Total) 17 1.30710
R squared = 1 - Residual 85 / Corrected $5 = .97639
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .237359602 .024306009 .185552570 .289166635
.A .023323244 .000740512 .021744879 .024901608
N 3.721848523 .483569795 2.691143904 4.752553142
Asymptotic Correlation Matrix of the Parameter Estimates
SR A N
SR 1.0000 .2700 .5130
A. .2700 1.0000 -.3104
N .5130 -.3104 1.0000
8.7. Cell 7
Nonlinear Regression Summary Statistics Dependent Variable PE_S7
Source DF Sum of Squares Mean Square
Regression 3 7.42435 2.47478
Residual 15 .05292 3.528294E-03
Uncorrected Total 18 7.47728
(Corrected Total) 17 1.12329
R Squared = 1 - Residual 58 / Corrected SS = .95288
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .221494731 .039091195 .138173821 .304815641
A- .020132438 .000656554 .018733025 .021531851

N 4.317164518 .764616195 2.687423678 5.946905358

151

.Asymptotic Correlation Matrix of the Parameter Estimates

SR A

SR 1.0000 .3580

A .3580 1.0000

N .6546 -.l672
8.8. P! All

N
.6546
-.1672
1.0000

Nonlinear Regression Summary Statistics

DE Sum of Squares

Source

Regression 3
Residual 123
Uncorrected Total 126
(Corrected Total) 125

R squared = 1 - Residual SS / Corrected SS =

Dependent Variable PE_S
Mean Square

58.08456 19.36152

.33164 2.696268E-03
58.41620
7.73521

Asymptotic
Std. Error
.014684055
.000383119

.147475130 2.828453167

.95713

.Asymptotic 95 8

Confidence Interval

Lower Upper
.200429752 .258562125
.021088691 .022605412

3.412289123

.Asymptotic Correlation Matrix of the Parameter Estimates

Parameter Estimate
SR .229495938
A .021847051
N 3.120371145
SR
SR 1.0000
A .2799
N .6713
C. Pooled SE and 88 data

A.
.2799
1.0000
-.2870

N
.6713
-.2870
1.0000

Nonlinear Regression Summary Statistics

Source
Regression
Residual
Uncorrected Total
(Corrected Total)

R squared = l - Residual 83 / Corrected SS =

DE Sum of Squares

3
168
171
170

Dependent variable S
Mean Square

74.52789 24.84263
.40717 2.423657E-03

74.93507

12.31172

.96693

.Asymptotic 95 8
Confidence Interval

Lower Upper
192365905 .235430670
021027223 .022352698

.110100837 2.803656840 3.238375722

Asymptotic
Parameter Estimate Std. Error
SR .213898288 .010906972
A. .021689960 .000335702
N 3.021016281
.Asymptotic Correlation Matrix of the Parameter Estimates
SR .A N
SR 1.0000 .1848 .6725
.A .1848 1.0000 -.3740
N .6725 -.3740 1.0000

152

8888 Result
Air-Oil System

A. Saturation Equilibrium

 

A.1. Cell 1
Nonlinear Regression Summary Statistics Dependent Variable 551
Source DF Sum of Squares Mean Square
Regression 3 3.14637 1.04879
Residual 6 1.654965E-03 2.758274E-04
Uncorrected Total 9 3.14803
(Corrected Total) 8 .73555
R squared = 1 - Residual 35 / Corrected $3 = .99775
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .216736677 .015467300 .178889556 .254583797
A. .062351208 .002559450 .056088460 .068613957
N 2.618260450 .140317205 2.274916619 2.961604281
.Asymptotic Correlation Matrix of the Parameter Estimates
SR A. N
SR 1.0000 -.O595 .7526
A. -.0595 1.0000 -.5531
N .7526 -.5531 1.0000
A12. Cell 2
Nonlinear Regression Summary Statistics Dependent Variable 552
Source DF Sum of Squares Mean Square
Regression 3 3.04630 1.01543
Residual 6 1.396263E-03 2.327105E-04
Uncorrected Total 9 3.04769
(Corrected Total) 8 .61374
R squared = 1 - Residual SS / Corrected 53 = .99772
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .208232505 .019519642 .160469662 .255995348
.A .079441789 .004012099 .069624537 .089259041
N 2.134933754 .105663076 1.876385521 2.393481986
.Asymptotic Correlation Matrix of the Parameter Estimates
SR A N
SR 1.0000 -.2380 .8413
A. -.2380 1.0000 -.6373
N .8413 -.6373 1.0000
Am3. Cell 3
Nonlinear Regression Summary Statistics Dependent variable SS3
Source DF Sum of Squares Mean Square
Regression 3 3.29934 1.09978
Residual 6 2.0448078-03 3.4080128-04
Uncorrected Total 9 3.30139
(Corrected Total) 8 .69184
R squared = 1 - Residual 55 / Corrected SS = .99704

Asymptotic 95 8
Asymptotic Confidence Interval

153

Parameter Estimate Std. Error Lower Upper
SR .237269389 .017900322 .193468879 .281069900
A .060570293 .002899210 .053476182 .067664404
N 2.574162336 .157311212 2.189235666 2.959089006
Asymptotic Correlation Matrix of the Parameter Estimates
SR A N
SR 1.0000 -.0662 .7579
A -.0662 1.0000 -.5611
N .7579 -.5611 1.0000
A.4. Cell 4
Nonlinear Regression Summary Statistics Dependent Variable SS4
Source DF Sum of Squares Mean Square
Regression 3 3.31058 1.10353
Residual 6 2.165306E-03 3.608844E-04
Uncorrected Total 9 3.31275
(Corrected Total) 8 .67805
R squared = 1 - Residual SS / Corrected $8 = .99681
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .256779702 .016891314 .215448146 .298111257
A .060096911 .002837410 .053154018 .067039803
N 2.721194994 .176107274 2.290276017 3.152113971
Asymptotic Correlation Matrix of the Parameter Estimates
SR .A N
SR 1.0000 -.0316 .7354
.A -.0316 1.0000 -.5389
N .7354 -.5389 1.0000
A15. SI ALL
Nonlinear Regression Summary Statistics Dependent variable SS_ALL
Source DF Sum of Squares Mean Square
Regression _ 3 12.79225 4.26408
Residual 33 .01760 5.334762E-04
Uncorrected Total 36 12.80986
(Corrected Total) 35 2.72315
R squared = 1 - Residual SS / Corrected SS = .99354
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .230852388 .011515977 .207422957 .254281820
A. .064636301 .001999256 .060568785 .068703818
N 2.495353457 .094577587 2.302933909 2.687773005

Asymptotic Correlation Matrix of the Parameter Estimates
SR A N

SR 1.0000 -.0941 .7733

A -.0941 1.0000 -.5712

N .7733 -.5712 1.0000

154

8. Presure'lquilibriumi

8.1. Cell 1
Nonlinear Regression Summary Statistics Dependent Variable PSl
Source DF Sum of Squares Mean Square
Regression 3 6.80869 2.26956
Residual 10 .02558 2.557503E-03
Uncorrected Total 13 6.83426
(Corrected Total) 12 .78736
R squared = 1 - Residual 55 / Corrected $5 = .96752
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .216224062 .064561620 .072371806 .360076315
A .067541880 .005856179 .054493500 .080590260
N 2.312055769 .301804838 1.639592684 2.984518853
Asymptotic Correlation Matrix of the Parameter Estimates
SR A N
SR 1.0000 .2226 .7835
A .2226 1.0000 -.2716
N .7835 -.2716 1.0000
8.2. Cell 2
Nonlinear Regression Summary Statistics Dependent variable P52
Source DF Sum of Squares Mean Square
Regression 3 6.40744 2.13581
Residual 10 .01025 1.024532E-03
Uncorrected Total 13 6.41768
(Corrected Total) 12 .97341
R squared = 1 - Residual SS / Corrected SS = .98947
.Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .222122383 .022709674 .171522076 .272722689
A. .052948823 .001498861 .049609152 .056288493
N 4.068471978 .344772143 3.300271770 4.836672186
Asymptotic Correlation Matrix of the Parameter Estimates
SR .A N
SR 1.0000 .3767 .4367
A. .3767 1.0000 -.3795
N .4367 -.3795 1.0000
8.3. Cell 3
Nonlinear Regression Summary Statistics Dependent variable P33
Source DE Sum of Squares Mean Square
Regression 3 6.54629 2.18276
Residual 10 5.794601E-03 5.794601E-04
Uncorrected Total 13 6.55409
(Corrected Total) 12 .85967

R squared = 1 - Residual 55 / Corrected 58 = .99326

155

Asymptotic 95 8

Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .240359759 .021526053 .192396725 .288322793
A .068707315 .001834661 .064619436 .072795193
N 3.177073165 .207895561 2.713852990 3.640293341
Asymptotic Correlation Matrix of the Parameter Estimates
SR A N
SR 1.0000 .3517 .6682
A .3517 1.0000 -.1998
N .6682 -.1998 1.0000
8.4. Cell 4
Nonlinear Regression Summary Statistics Dependent variable PS4
Source DE Sum of Squares Mean Square
Regression 3 6.27705 2.09235
Residual 10 2.939531E-03 2.939531E-04
Uncorrected Total 13 6.27999
(Corrected Total) 12 .82741
R squared = 1 - Residual SS / Corrected 88 = .99645
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .270039190 .018093034 .229725398 .310352982
.A .058346729 .001846515 .054232436 .062461022
N 2.464993945 .122208989 2.192695348 2.737292541
Asymptotic Correlation Matrix of the Parameter Estimates
SR .A N
SR 1.0000 .0668 .7907
A. .0668 1.0000 -.4043
N .7907 -.4043 1.0000
8.5. P! All
Nonlinear Regression Summary Statistics Dependent variable PséALL
Source DF Sum of Squares Mean Square
Regression 3 25.93174 8.64391
Residual 49 .15428 3.1486143-03
Uncorrected Total 52 26.08602
(Corrected Total) 51 3.45833
R squared = 1 - Residual SS / Corrected SS = .95539
Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .256510471 .024897608 .206476855 .306544088
.A .063704804 .002514772 .058651180 .068758429
N 2.797045959 .206964647 2.381134929 3.212956988
Asymptotic Correlation Matrix of the Parameter Estimates
SR A. N
SR 1.0000 .2279 .6752

.A .2279 1.0000 -.3471
N .6752 -.3471 1.0000

156

C. Pooled 83 and PB date

Nonlinear Regression Summary Statistics Dependent Variable S_TOTAL
Source DF Sum of Squares Mean Square
Regression 3 38.72051 12.90684
Residual 85 .17537 2.063226E-03
Uncorrected Total 88 38.89588
(Corrected Total) 87 6.54283
R squared = 1 - Residual 55 / Corrected $5 = .97320

Asymptotic 95 8
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
SR .245726009 .014516142 .216864030 .274587988
A .064034385 .001806441 .060442697 .067626074
N 2.659540349 .123329373 2.414328515 2.904752183
Asymptotic Correlation Matrix of the Parameter Estimates
SR A. N
SR 1.0000 .0716 .7271

A .0716 1.0000 -.4429
N .7271 -.4429 1.0000

157

D. INTERFACIAL TENSION DATA

D.1.
D.2.
D.3.
D4.
D5.
D.6.

Air-Water System - Metea Soil
Oil-wter System - Metea Soil
Air-Oil System - Metea Soil
Air-Water System - Ottawa Sand
Oil-Water System - Ottawa Sand
Air-Oil System - Ottawa Sand

158

Air. (-)
Water: air-water experiment (ME 1 14 and ME 1 15)
me114#1la me 114#2/a
Density= 0.974 0.9836
Mag Coef= 49.325 50
de 05 y de 08 7

109.3 59.8 71.85 111.6 61.8 71.34

109.1 60.2 70.02 111.4 61.8 70.75

109.1 60 70.63 1 1 1.5 62 70.45

110 60.6 71.48 111.1 61.7 70.17

109.9 60.7 70.87 110.9 62 68.72

109.9 60.5 71.49 110.9 61.6 69.89

110 61.5 68.79 111.3 61.9 70.16

110.2 61.3 69.95 111.1 62.1 69.00

110.2 61.6 69.07 111.1 61.9 69.59

110 61 70.26 110.7 61.2 70.50

110.1 61 70.56 110.7 61.1 70.80

110 61 .1 69.96 110.9 61.3 70.78

109.8 61 69.68 1 1 1 61.8 69.59

109.8 60.9 69.98 111 62 69.01

109.8 60.8 70.28 1 11 61.8 69.59

109.9 60.9 70.27 111 61.8 69.59

109.7 60.7 70.28 110.9 61.8 69.30

109.9 60.5 71.49 111 61.8 69.59

AVG 70.38 69.93

STDEV 0.802 0.707

me 114113/a me 114 #4Ia
Density= 0.9818 0.9766
Mag Coeff- 50.2 49.975
’ de 05 y de ds 1

110 61 68.38 111.1 62.5 67.44

109.9 61.4 66.94 111.1 62.2 68.29

109.9 61.2 67.52 111 62.1 68.30

110.2 61.4 67.79 111.1 62.6 67.16

110.1 61.4 67.51 111.1 62.7 66.88

110 61.1 68.09 110.9 62.6 66.61

110 61.4 67.22 110.9 61.9 68.59

109.5 61 66.96 110.8 61.7 68.88

110 61.3 67.51 111 61.8 69.16

110.2 61 68.95 111 62 68.58

110.2 61.5 67.50 110.9 61.8 68.88

110.1 61.5 67.22 110.9 61.9 68.59

109.8 61.1 67.52 110.3 62.5 65.23

109.8 61 67.81 110.3 62.4 65.51

109.5 61.4 65.83 110.4 62.4 65.78

110 61.8 66.10 110.8 62.2 67.45

109.9 61.8 65.82 110.9 62.2 67.73

110 61.8 66.10 110.9 62.3 67.45

AVG 67.26 67.58

STDEV 0.841 1.171

 

 

159

Air. (-)
Water. air-water experiment (ME 114 and ME 115)
me115#1la me115#2la
Density= 0.9576 0.9738
Mag C0ef= 50 49.875
de ds 1 de ds 7
110.6 60.9 69.23 110 61 68.71
110.5 61.1 68.35 110 60.9 69.00
110.5 61.1 68. 35 110 61 68.71
110.9 61 69. 80 111.1 61.8 69.53
110.9 61.1 69.50 111.2 61.6 70.41
110.9 61.2 69.21 111.1 61.6 70.12
110.6 61 68.93 110.7 61.2 70.14
110.6 61.2 68.35 110.6 61.4 69.26
110.7 61.1 68.93 110.7 61.2 70.14
110.9 61.4 68.62 111 62 68. 66
111 61.6 68.32 110.9 62.2 67. 81
111 61.7 68.04 111 62 68.66
110.6 61.1 68.64 110.9 61.9 68.67
110.7 61 69. 22 110.9 61.8 68.96
110.6 61.2 68. 35 110.9 62 68. 38
110.7 61.1 68. 93 110.6 61.7 68.39
1 10.8 61.1 69.21 1 10.4 62 66.97
110.7 61.3 68.34 110.6 62 67.53
AVG 68.80 68.89
STDEV 0.478 0.912
AVG 68.65
me 11583/a me 115#4/a STDEV 1.51
Density= 0.9721 0.9804 N 144.00
Mag Coef= 46.66 50.325 t-95 3.411228
09 ds 7 06 ds 7
103.1 57.7 67.22 110.5 61.9 66.78
103.1 57.8 66. 92 110.6 62 66.78
103.1 58 66. 32 110.5 61.9 66.78
103.2 57.8 67. 22 112.1 62. 5 69.58
103.2 57.7 67.52 112.2 62. 5 69.87
103.3 57.8 67.52 112.2 62. 5 69.87
103.1 57.6 67.52 112.1 62. 5 69.58
103.1 57.8 66.92 112.3 62 .6 69.86
103.2 57.6 67.83 1 12.1 62 71.05
103.1 57.9 66.62 112.1 62. 7 69. 00
103.1 57.9 66.62 112.1 62.9 68.43
103.1 58 66.32 112.2 62.8 69. 00
103 57.9 66.32 1 12.4 62.5 70. 44
103 57.9 66.32 112.3 62.5 70.15
103 57.7 66.92 1 12.3 62.7 69.57
103.1 57.8 66.92 111.9 62.1 70.17
103 57.8 66.62 1 12 61.6 71.96
103 57.8 66. 62 111.9 61.8 71.06
AVG 66.90 69.44
STDEV 0.465 1.431

OiI: Air-oil experiment (ME 124)

Water oil-water experiment (ME 125)

Density=
dag Coef=

AVG

STDEV

Density=
Jag Coef=

#1

0.6028

43.5
de
93.5
93.5
93.5
91.5
91.5
91.6
96.9
96.9
96.9

#3

0.6088
42.817

de
93.5
93.5
93.5
90.5
90.7
90.5
92.7
92.6
92.5

AVG

53
53.1
53
52.5
52.5
52.2
54.5
55
54.7

52.1
52.3
52.4
51 .4
51 .2
51 .2
52.9
52.8
52.8

STDEV

38.15
37.98
38.15
35.40
35.40
36.11
41.82
40.84
41.42
38.38
2.365

41.45
41.04
40.83
38.95
37.71
37.33
38.29
38.67
38.10
38.93
1.819

160

0.5988

41 .75
de
91
91 .1
91
89.2
89.1
89.3
93.5
93.7
93.5

0.5918

42.95

93.4
93.2
93.3

94.1
93.1

93

93
95.1
95.1
95.1
95.7
95.9
95.8
96.1
96.2
98.2

52
51.7
51.9
51.2
51.5
51.2
52.6
52.2
52.9

ds

AVG

54.1

53.9

52.8
52.7
52.8
54.3
54.1
54.5
54.5
54.5
53.9
55.1
54.8
55.1

STDEV

AVG

STDEV

{-95

38.16
38.94
38.35
38.24
35.51
36.43
41.98
43.23
41.35
38.91
2.575

7
38.22
38.04
38.39
37.49
37 .49
37.87
38.03
38.03
37.84
38.99
39.38
38.81
39.75
40.14
41 .1 1
39.38
40.14
39.57
38.48
1 .454

38.82
1 .97815
45

4.590717

24.01
24. 1 8
24.85
25.25
25.25
25.25
25.25
25.73
25.59
25.02
0.571

25.53
24.85
25.53
25.17
25.51
25.15
25.56
24.84
25.03
25.22

Air. (-)
Oil: air-oil experiment (ME 118)
me 118 0a It 1
Density= 1.598
dag Coef= 46.824
de ds

60.1 39.8
60.1 39.7
60.1 39.4
60.5 39.5
60.5 39.5
60.5 39.5
60.5 39.5
60.4 39.1
60.5 39.3

AVG

STDEV

me 118 0a # 3
Density= 1.5746
wag Coef= 46.396
de ds

60.6 39.5
60.4 39.8
60.6 39.5
60.5 39.6
60.5 39.4
60.4 39.5
60.7 39.6
60.5 39.8
60.6 39.8

AVG

STDEV

0.317

161

me 118 0a # 2
1.5982
46.432
de ds
60.9 39.9
60.9 39.9
60.9 39.7
60.7 40.1
60.9 39.7
60.9 39.9
60.9 40.0
60.9 40.0
60.9 39.9
me 116 0a # 4
1.5867
46.204
de 05
60.9 40.8
61.1 40.7
61.1 40.8
61.0 40.1
61.1 39.9
61.9 402
61.0 41.0
61.0 41.0
61.1 40.9
AVG
STDEV
N

1-95

25.78
25.78
28.13
25.06
28.13
25.78
25.82
25.82
25.78
25.74
0.297

24.38
24.92
24.78
25.71
28.25
27.33
24.28
24.28
24.60
25.18
0.999

25.29
0.89
27.00

1 .850298

Air :

Water. Oil-Water experiment (OS 121 and OS 122)
as 121 ow #1Ia

Density=
dag Coef=

AVG
STDEV

Density=

dag Coef= 46.275

AVG
STDEV

Density=
dag Coef=

AVG
STDEV

(-)

0.9844
45.975
de ds
102.8 56.1
102.6 56.6
102.7 56.4
102.6 56.9
102.6 56.5
102.6 56.9
103.0 57.0
102.9 57.2
102.9 56.9
as 121 ow #318
0.9769
de ds
103.3 57.3
103.5 56.8
103.5 57.0
103.7 57.3
103.5 57.5
103.6 57.1
103.4 57.1
103.3 57.7
103.3 57.1
as 122 CW #1Ia
0.9824
46.2
de 05
103.0 57.0
103.0 57.1
103.1 56.9
101.6 57.2
101.7 56.9
101.9 56.9
103.0 57.5
102.9 57.5
103.0 57.4

74.43
72.09
73.08
71 .10
72.42
71 .10
72.08
71 .08
72.06
72.18
1.028

70.58
72.84
72.17
71.83
70.55
72.18
71.53
89.30
71.21
71.35
1.017

71 .21
70.89
71 .88
88.25
87.47
88.09
89.81
89.30
89.93
89.40
1 .738

162

05 121 ow #2/a
0.9773
46
de ds
102.8 56.9
102.9 56.7
102.9 56.8
103.0 56.6
102.9 56.6
103.1 56.1
102.9 57.0
102.9 56.9
103.0 57.0
as 121 ow #4Ia
0.9824
46.325
de ds
103.8 57.2
103.9 57.1
103.9 56.9
103.4 57.5
103.5 57.5
103.2 57.4
103.8 57.9
103.9 57.9
104.0 57.6
as 122 0w # 21a
0.9832
46.25
de ds
103.1 56.7
103.1 57.0
103.1 56.9
103.1 56.7
103.0 56.6
103.0 57.0
103.0 57.0
103.1 56.8
103.1 56.9

71.15
72.12
71.79
72.78
72.45
74.81
71.14
71.46
71.46
72.13
1.089

72.73
73.38
74.06
70.48
70.79
70.17
70.48
70.77
72.06
71 .86
1 .358

72.42
71.43
71.78
72.42
72.43
71.1 1
71.11
72.09
71.78
71.84
0.507

Air:

(-)

163

Water. Oil-Water experiment (OS 121 and OS 122)

Density=
Mag Coef=

AVG
STDEV

es 122 ow # 3/a
0.979
47.175
de ds
106.0 58.5
106.0 58.8
106.0 58.7
106.0 59.0
106.0 58.9
105.9 58.9
106.0 59.0
106.0 58.7
105.9 59.0

7
72.60

71.64
71.96
71.01
71.32
71.01
71.01
71.96
70.70
71.47
0.580

as 122 0w #4/a
0.9816
47.675
de ds
105.9 58.0
105.9 57.9
105.8 58.0
105.9 57.5
105.7 57.9
105.9 57.8
105.9 58.0
105.9 57.8
105.9 . 58.1
AVG
STDEV
N
t-95

7
72.57

72.89
72.25
74.22
72.28
73.22
72.57
73.22
72.24
72.83
0.81 3

72.14

1 .432
72

1 .881 245

164

Oil : Air-oil experiment (OS 123)
Water: oil-water experiment (OS 12)

-os1220w#1 os1220w#2
Density= 0.61 79 0.6048
.439 Coef= 40.825 44.1
de ds 1 de 05 y
83. 1 46.0 37.31 87.9 46.8 38.74
83.0 46.0 37.10 87.8 46.8 38.54
83-1 45.9 37.52 87.8 47.0 38.12
84.2 46.2 39.19 87.2 47.0 36.93
84.5 46.2 39.84 87.1 46.8 37.15
84.2 46.0 39.64 87.1 47.0 36.74
82.8 46.0 36.69 88.8 48.2 37.61
82.8 46.2 36.28 88.8 48.0 38.02
82.7 45.9 36.70 88.8 48.2 37.61
AVG 37.81 37.72
STDEV 1.292 0.658
es 122 ow #3 as 122 ow #4
Density= 0.6133 0.5827
hag Coef= 40.1 5 43.2
de 05 y de ds 7
88.1 50.9 38.49 91.8 50.1 39.79
88.1 51.6 37.14 91.9 50.1 39.99
88.0 51.0 38.09 92.0 50.1 40.19
86.1 50.1 36.08 89.6 48.9 37.90
86.1 50.0 36.27 89.3 48.7 37.72
86.0 50.0 36.08 89.4 49.0 37.32
87.9 51.0 37.89 91.9 50.1 39.99
87 .9 51.0 37.89 91.9 50.0 40.20
88.0 50.9 38.29 91.9 50.0 40.20
AVG 37.36 39.25
STDEV 0.928 1.152
AVG 38.03
STDEV 1.281291
N 36

t-95 3.00074

25.49
25.68
25.85
25.54
28.06
25.54
25.51
26.19
26.38
25.80
0.314

28.83
27.84
27. 88
26.09
28. 82
28.28
26.97
28. 80
28.97
27.14

Air : (-)
Oil : air-oil experiment (OS 123)
as 123 so #1 la
Density= 1.6013
JIag Coef= 47.775
de ds

62.0 40. 5
62.1 40. 5
62.1 40. 4
622 40.7
62.3 40.5
62.2 40.7
62.1 40.6
62.1 402
62.2 40.2

AVG

STDEV

es 123 so # 3Ia
Density= 1.5913
illag Coef= 46.3
de ds

62.9 40.5
62.6 40. 7
62.7 40. 8
61.9 40. 9
62.0 40. 7
62.0 40. 9
62.0 40 .5
62.0 40. 6
62.0 40. 5

AVG

STDEV

0.825

165

05 123 a0 #2/a
1.5851
47.275
de ds
62.6 40 9
62.7 40. 9
62.7 40. 9
62.1 40.6
62.2 40. 4
62.1 40. 2
62.8 40.8
62.7 40.9
62.6 40.7
as 123 0a #4/a
1.5647
4625
de ds
61.9 40.1
61.8 40.0
61.8 40.3
61.5 40. 0
61.6 40.1
61.5 40.0
61.1 40.3
61.1 40.1
61.1 40.1
AVG
STDEV
N

1-95

26.26
28. 45
28.45
25.79
26.32
28.47
28.82
26.45
28.80
26. 40
0.262

27.07
27.05
28.52
26.45
28.47
26.45
25.18
25.50
25.50
26.24
0.853

28.40
0.75
38.00

1.7823

166

E. PRESSURE-SATURATION DATA OBTAINED USING
PRESSURE EQUILIBRIUM METHOD

E.1. Air-Water Data - Metea Soil.
E2. Oil-Water Data - Metea Soil.
E.3. Air-Oil Data - Metea Soil.

B.4. Air-Water Data - Ottawa Sand.
E.5. Oil-Water Data - Ottawa Sand
E.6. Air-Oil Data - Ottawa Sand.

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