Advances in the Theory of Capillarity in Porous Media

Advances in the Theory of Capillarity in Porous Media
GEOLOGICA ULTRAIECTINA
Mededeling van de
Faculteit Geowetenschappen
Universiteit Utrecht
No.314
Advances in the Theory of
Capillarity
in Porous Media
Simona Bottero
Advances in the Theory of
Capillarity
in Porous Media
Vooruitgang in de Theorie van Capillariteit
in Poreuze Media
(met een samenvatting in het Nederlands)
Proefschrift
ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van
de rector magnificus, prof.dr. J.C. Stoof, ingevolge het besluit van het college
voor promoties in het openbaar te verdedigen op maandag 9 november 2009 des
ochtends te 10.30 uur
door
Simona Bottero
geboren te Cagliari, Italië
Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. S.M.Hassanizadeh
Samenstelling leescommissie:
Prof.dr. R.Helmig
Prof.dr. T.H.Illangasekare
Prof.dr. L.J.Pyrak-Nolte
Dr. D.Wildenschild
Dr. C.Berentsen
Stuttgart University
Colorado School of Mines
Purdue University
Oregon State University
Delft University of Technology
Printed by Wöhrmann Print Service, Zutphen
ISBN 978-90-5744-175-7
to my parents
Marta and Francesco
Contents
Table of contents
iii
List of Figures
vii
List of Tables
xvi
1 Introduction
1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 General theory of two-phase flow
2.1 Interfacial tension and wettability . . . . . . . . . . . . . . .
2.2 Concept of capillary pressure at the microscale . . . . . . .
2.3 Capillary pressure at the macroscale . . . . . . . . . . . . .
2.4 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 The relative permeability-saturation relationship . . . . . .
2.6 Two-phase flow equations at the macroscale . . . . . . . . .
2.6.1 Darcy’s law for two phase flow . . . . . . . . . . . .
2.6.2 Mass conservation . . . . . . . . . . . . . . . . . . .
2.7 Two-phase flow theory after Hassanizadeh and Gray (1990)
3 What is the correct definition of
3.1 Introduction . . . . . . . . . . .
3.2 Averaging operators . . . . . .
3.3 Model description . . . . . . . .
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CONTENTS
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5 Local and average capillary pressure-saturation relationships
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Steady-state experiments . . . . . . . . . . . . . . . . . . . . . .
5.3 Calibration procedure . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Results of primary drainage experiments . . . . . . . . . .
5.4.2 Results of main drainage and imbibition experiments . . .
5.5 Average capillary pressure-saturation relationship . . . . . . . . .
5.5.1 Averaging operators . . . . . . . . . . . . . . . . . . . . .
5.6 Reference average capillary pressure . . . . . . . . . . . . . . . .
5.6.1 Results and discussion . . . . . . . . . . . . . . . . . . . .
5.7 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . .
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6 Non-equilibrium two-phase flow experiments
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Overview of non-equilibrium experiments . . . . . . .
6.3 Non-equilibrium experiments . . . . . . . . . . . . . .
6.3.1 Results from primary drainage experiments . .
6.4 The average non-equilibrium phase pressure difference
89
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99
3.4
3.5
3.6
3.7
3.8
3.9
3.3.1 Pressure distribution across the domain .
Results and discussion . . . . . . . . . . . . . . .
3.4.1 The correct average pressure . . . . . . .
Average pressure during primary drainage . . . .
Average pressure during main drainage . . . . . .
Pn above the infiltration front for main drainage
When is the use of a correct operator important?
Summary and conclusions . . . . . . . . . . . . .
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4 Materials and Methods
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Experimental set-up . . . . . . . . . . . . . . . . . . . .
4.3 Pore Pressure transducers . . . . . . . . . . . . . . . . .
4.3.1 Procedure for differential phase pressure reading
4.3.2 Calibration and absolute error . . . . . . . . . .
4.4 Time Domain Reflectometry . . . . . . . . . . . . . . . .
4.4.1 Calibration of TDR system . . . . . . . . . . . .
4.5 Sand column preparation . . . . . . . . . . . . . . . . .
4.6 Obtaining saturation column from differential pressures
4.7 Experimental issues and remarks . . . . . . . . . . . . .
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CONTENTS
6.5
6.6
6.7
6.8
The non-equilibrium coefficient τ at the local scale . .
The non-equilibrium coefficient τ at the column scale .
Discussions . . . . . . . . . . . . . . . . . . . . . . . .
Summary and conclusions . . . . . . . . . . . . . . . .
vii
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103
109
114
119
7 Numerical simulation of non-equilibrium two-phase experiments123
7.1 Description of the numerical code . . . . . . . . . . . . . . . . . . 123
7.2 Conceptual model . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2.1 Domain description and medium-fluids properties . . . . . 126
7.2.2 Boundary and initial conditions . . . . . . . . . . . . . . . 127
7.3 Averaging simulated fluid pressures . . . . . . . . . . . . . . . . . 129
7.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 131
7.4.1 Simulation of drainage experiments at 20 kPa . . . . . . . 132
7.4.2 Simulation of drainage experiments at 30 kPa . . . . . . . 136
7.4.3 Simulation of drainage experiments at 35 kPa . . . . . . . 141
7.4.4 Simulation of drainage experiments at 38 kPa . . . . . . . 146
7.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . 146
8 Pc − S − awn relationship in a 2D micromodel
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 2D Micromodel Experiments . . . . . . . . . . . . . . . . . .
8.2.1 Experimental set-up . . . . . . . . . . . . . . . . . . .
8.2.2 Fabrication of the micromodel . . . . . . . . . . . . .
8.2.3 Experimental procedure . . . . . . . . . . . . . . . . .
8.3 Data image analysis . . . . . . . . . . . . . . . . . . . . . . .
8.4 Pc − sw − awn surfaces . . . . . . . . . . . . . . . . . . . . . .
8.5 Results and discussions . . . . . . . . . . . . . . . . . . . . .
8.5.1 Equilibrium experiments . . . . . . . . . . . . . . . . .
8.5.2 Non-equilibrium drainage and imbibition experiments
8.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . .
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9 Summary and Conclusions
181
Samenvatting
187
Appendix
193
Acknowledgements
197
About the Author
201
List of Figures
2.1
Two immiscible fluids is a capillary tube. . . . . . . . . . . . . .
3.1
a) schematic representation of manometer used to measure pressure at a given elevation z1 in a porous medium column fully
saturated with water (drawing is not to scale); b) schematic representation of pore distribution at the manometer opening, showing
larger pores in the lower half of the opening; c) pressure distribution across the manometer opening. . . . . . . . . . . . . . . . .
schematic pressure distribution for the wetting and nonwetting
phase in a vertical one-dimensional homogeneous domain at static
equilibrium. The nonwetting phase has a higher density than the
wetting phase, and infiltrates the domain from below. a) The
nonwetting phase has infiltrated the domain up to zf ; b) the
nonwetting phase has infiltrated the domain completely. . . . . .
Pressure and saturation distributions used to obtain average pressures for ∆P = 0.25 and ρ = 1.6; a) potential-based average pressure; b) intrinsic phase-volume average pressure; c) simple phase
average pressure; d) simple average pressure. . . . . . . . . . . .
Pressure and saturation distributions used to obtain average pressures for ∆P = 0.25 and ρ = 1.6; a) centroid-corrected phase
average pressure; b) saturation distribution. . . . . . . . . . . . .
Average pressures as a function of average saturation for primary
drainage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Centroid of the domain and centroids of the phases as a function
of average water saturation < Sw > during primary drainage. .
3.2
3.3
3.4
3.5
3.6
ix
7
21
26
30
31
32
33
x
LIST OF FIGURES
3.7
3.8
3.9
3.10
3.11
3.12
3.13
a) Choice of pressure distribution above infiltration front. The
1
shaded area corresponds to Pn0
, which can be anything between
1
Pw and Pw + Pd . However, here we assume Pn0
= Pw + Pd above
the infiltration front; b) Corresponding saturation distribution. .
Average pressures as a function of average, for main drainage,
1
assuming Pn0
(Pn = Pw + Pd above zf ). . . . . . . . . . . . . . .
Average pressures as a function of average saturation, for main
2
drainage, assuming Pn0
. . . . . . . . . . . . . . . . . . . . . . . .
Centroid of the domain and centroids of the phases as a function
of average water saturation < Sw > during main drainage. . . . .
a) < Pn > − < Pw > vs. < Sw > for the main drainage process,
using nonwetting phase pressure distribution , and a dimensionless
entry pressure of 1.53; b) the same, but with a dimensionless entry
pressure of 0.06. . . . . . . . . . . . . . . . . . . . . . . . . . . .
a) Pressure distribution and average pressures for two identical
domains placed on top of each other; b) corresponding saturation
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
a) values for the different nonwetting phase average pressures as
a function of Ng. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2
4.3
Experimental setup (not to scale). . . . . . . . . . . . . . . . . .
Sand column set-up. . . . . . . . . . . . . . . . . . . . . . . . . .
On the left: fluid displacement throughout drainage experiment;
on the right: imbibition experiment. . . . . . . . . . . . . . . . .
4.4 On top: holder with circular and radial channels; Below: stainlesssteel porous plate. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 A, Pore pressure transducer; B, hydrophillic or hydrophobic filters; C, hydrophillic or hydrophobic membranes; D, transducer
case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Hydrophobic and hydrophillic filters and membranes after experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Calibration curves of pore pressure transducers. . . . . . . . . . .
4.8 Time Domain Reflectometry probe. . . . . . . . . . . . . . . . . .
4.9 Inflowing burette, dP1 and dP2 are differential pressures. . . . .
4.10 Original experimental setup. . . . . . . . . . . . . . . . . . . . . .
5.1
Measured nonwetting and wetting phase pressures versus equilibrium steps along with the external pressure throughout primary
drainage experiment. . . . . . . . . . . . . . . . . . . . . . . . . .
36
37
37
38
39
41
43
48
49
49
50
53
54
55
56
59
61
67
LIST OF FIGURES
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
xi
Measured local water saturations and average water saturation
versus equilibrium steps during primary drainage experiment. . . 67
Local capillary pressure versus local water saturation at different
elevations along with the external pressure versus average saturation throughout primary drainage. . . . . . . . . . . . . . . . . . 68
Measured wetting and nonwetting pressures versus equilibrium
steps along with the external pressure throughout main drainage
experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Measured water saturation versus equilibrium steps in main drainage. 69
Local capillary pressure versus water saturation along with the
external pressure versus average water saturation in main drainage. 70
Measured local wetting and nonwetting phase pressures along
with the external pressure versus equilibrium steps throughout
main imbibition experiment. . . . . . . . . . . . . . . . . . . . . 71
Measured local water saturation throughout main imbibition. . . 71
Local capillary pressure versus water saturation along with the external pressure versus average saturation during main imbibition
experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Capillary pressure-saturation curves at elevation z1 , z2 , and z3
along with the Pc − Sw curve fitted by van Genuchten model:
a) Primary Drainage; b) Main Drainage; c) Main Imbibition; d)
comparison between Pc −Sw curves during primary drainage, main
drainage and main imbibition fitted by Van Genuchten model. . 74
Averaged nonwetting and wetting phase pressures versus equilibrium steps during primary drainage . . . . . . . . . . . . . . . . . 80
Centroid of the average domain and centroids of the wetting and
nonwetting phases versus equilibrium steps during primary drainage. 80
Average Pc − Sw curves along with the local one obtained by
fitting Pc − Sw data by Van Genuchten model. All curves refer to
primary drainage. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Averaged nonwetting and wetting phase pressures versus equilibrium steps during main drainage. . . . . . . . . . . . . . . . . . . 82
Centroid of the average domain and centroids of the wetting and
nonwetting phases versus equilibrium steps throughout main drainage. 82
Capillary pressure-saturation curves based on different averaging
operators along with the reference curve and the local one. These
refer to main drainage. . . . . . . . . . . . . . . . . . . . . . . . . 83
Average nonwetting and wetting phase pressures during main imbibition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
xii
LIST OF FIGURES
5.18 Centroid of the average domain along with the centroid of the wetting and nonwetting phase versus equilibrium steps during main
imbibition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.19 Average capillary pressure-saturation curves obtained by different
averaging operators along with the reference curve and with the
local Pc − Sw . These regard main imbibition process. . . . . . . .
6.1
6.2
6.3
6.4
84
85
Non-equilibrium experiment results at imposed pressure of 20
kPa.(a) and (b) wetting and nonwetting phase pressures versus
early and later time respectively; (c) and (d) local water saturation versus time; (e) and (f) local pressure difference versus time
at early and later time. . . . . . . . . . . . . . . . . . . . . . . .
95
Non-equilibrium experiments results at imposed pressure of 30
kPa. (a) and (b) wetting and nonwetting phase pressures versus
early and later time; (c) and (d) local water saturation versus
time; (e) and (f) pressure difference versus early and later time
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
Non-equilibrium experiment results at imposed pressure of 35
kPa. (a) and (b) wetting and nonwetting phase pressures versus time; (c) and (d) local water saturation versus time; (e) and
(f) local pressure difference versus time. . . . . . . . . . . . . . .
97
Non-equilibrium experiment results at imposed pressure of 38
kPa. (a) and (b) wetting and non-wetting phase pressures versus time; (c) and (d) local water saturation versus time; (e) and
(f) local capillary pressure. . . . . . . . . . . . . . . . . . . . . .
98
6.5
Local non-equilibrium phase pressure difference-saturation curves
at applied pressure of 20 kPa, 30 kPa, 35 kPa, 38 kPa at elevation
z1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.6
Local non-equilibrium phase pressure differences-saturation curves
at applied pressure of 20 kPa, 30 kPa, 35 kPa, 38 kPa at elevation
z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.7
Local non-equilibrium phase pressure differences-saturation curves
at applied pressure of 20 kPa, 30 kPa, 35 kPa, 38 kPa at elevation
z3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.8
Average non-equilibrium pressure differences versus average wetting saturation; corresponding local curve at three elevations z1 ,
z2 , and z3 for injection pressure of 20 kPa, and the average capillary pressure-saturation curve. . . . . . . . . . . . . . . . . . . . 102
LIST OF FIGURES
6.9
xiii
Average non-equilibrium pressure differences versus average wetting saturation; corresponding local curve at three elevations z1 ,
z2 , and z3 for injection pressure of 30 kPa, and the average capillary pressure-saturation curve. . . . . . . . . . . . . . . . . . . . 103
6.10 Average non-equilibrium pressure differences versus average wetting saturation; corresponding local curve at three elevations z1 ,
z2 , and z3 for injection pressure of 35 kPa, and the average capillary pressure-saturation curve. . . . . . . . . . . . . . . . . . . . 104
6.11 Average non-equilibrium pressure differences versus average wetting saturation; corresponding local curve at three elevations z1 ,
z2 , and z3 for injection pressure of 38 kPa, and the average capillary pressure-saturation curve. . . . . . . . . . . . . . . . . . . . 105
6.12 a) (Pn −Pw )−Pc versus water saturation; b) ∂Sw /∂t versus water
saturation at elevation z1 for injection pressure 20 kPa, 30 kPa,
35 kPa and 38 kPa. . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.13 (Pn − Pw ) − Pc versus ∂Sw /∂t at various water saturation. The
slope of each curve represent the material coefficient τ . . . . . . . 106
6.14 Non-equilibrium coefficient τ at local scale versus water saturation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.15 (Pn − Pw ) − Pc versus ∂Sw /∂t at various water saturation. The
slope of each curve represents the material coefficient τ . The
regression curve is not forced through the origin as in Equation 6.8.108
6.16 The coefficient τ versus water saturation according to Equation 6.8.108
6.17 The non-equilibrium coefficient τ versus local water saturation for
different injection pressures, by method three. . . . . . . . . . . . 110
6.18 Fitted average capillary pressure curve by van Genughten model. 110
6.19 Difference between the average phase pressure difference, [Pn ] −
[Pw ], and the capillary pressure versus the rate of change of the
average water saturation, ∂ < Sw > /∂t. In the same figure the
regression curves for various water saturation are plotted. The
slope of each curve represents the non-equilibrium coefficient, [τ ]
at the column scale. . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.20 Difference between the average phase pressure difference < Pn >i
− < Pw >i and the capillary pressure versus the rate of change
of the average water saturation. In the same figure the regression curves for various water saturation are plotted. The slope of
each curve represents the non-equilibrium coefficient at the column scale < τ >i . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
xiv
LIST OF FIGURES
6.21 Average non-equilibrium coefficients [τ ] and < τ > based on the
centroid-corrected and intrinsic phase average pressure respectively.113
6.22 Measured versus predicted water and oil pressure head from. The
solid symbol represents the water phase while the open symbol
the oil phase pressure. The continuous and dashed lines represent
the results of the numerical simulations Lenhard et al., 1988. . . 115
6.23 Measured water saturation versus time from Lenhard et al., 1988. 115
7.1
Conceptual model . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2
Local capillary pressure-saturation fitted by Brooks-Corey model. 129
7.3
Simulated average pressure differences using intrinsic-phase averaging and centroid-corrected averaging operators. The average is
performed over an area of 1 cm in diameter. . . . . . . . . . . . 130
7.4
Differences in results obtained by intrinsic-phase averaging and
the centroid-corrected averaging operators. . . . . . . . . . . . . . 131
7.5
Comparison of simulated and measured curves versus time at elevation z1 for the injection pressure of 20 kPa: a) phase pressures;
b) water saturation; c) non-equilibrium pressure difference. . . . 133
7.6
Comparison of simulated and measured curves versus time at elevation z2 for the injection pressure of 20 kPa a) phase pressures;
b) water saturation; c) non-equilibrium pressure difference. . . . 134
7.7
Comparison of simulated and measured curves versus time at elevation z3 for the injection pressure of 20 kPa a) phase pressures;
b) water saturation; c) non-equilibrium pressure difference. . . . 135
7.8
Measured and simulated column-scale average non-wetting phase
saturation versus time at injection pressure of 20 kPa. . . . . . . 137
7.9
Comparison of simulated and measured curves versus time at elevation z1 for the injection pressure of 30 kPa a) phase pressures;
b) water saturation; c) non-equilibrium pressure difference . . . 138
7.10 Comparison of simulated and measured curves versus time at elevation z2 for the injection pressure of 30 kPa a) phase pressures;
b) water saturation; c) non-equilibrium pressure difference . . . 139
7.11 Comparison of simulated and measured curves versus time at elevation z3 for the injection pressure of 30 kPa a) phase pressures;
b) water saturation; c) non-equilibrium pressure difference . . . . 140
7.12 Measured and simulated average PCE saturation versus time at
injection pressure of 30 kPa. . . . . . . . . . . . . . . . . . . . . 141
LIST OF FIGURES
xv
7.13 Comparison of simulated and measured curves versus time at elevation z1 for the injection pressure of 35 kPa: a) phase pressures;
b) water saturation; c) non-equilibrium pressure difference. . . . 142
7.14 Comparison of simulated and measured curves versus time at elevation z2 for the injection pressure of 35 kPa: a) phase pressures;
b) water saturation; c) non-equilibrium pressure difference. . . . 143
7.15 Comparison of simulated and measured curves versus time at elevation z3 for the injection pressure of 35 kPa: a) phase pressures;
b) water saturation; c) non-equilibrium pressure difference. . . . 144
7.16 Measured and simulated average PCE saturation versus time at
injection pressure of 35 kPa. . . . . . . . . . . . . . . . . . . . . 145
7.17 Comparison of simulated and measured curves versus time at elevation z1 for the injection pressure of 38 kPa: a) phase pressures;
b) water saturation; c) non-equilibrium pressure difference. . . . 147
7.18 Comparison of simulated and measured curves versus time at elevation z2 for the injection pressure of 38 kPa: a) phase pressures;
b) water saturation; c) non-equilibrium pressure difference. . . . 148
7.19 Comparison of simulated and measured curves versus time at elevation z3 for the injection pressure of 38 kPa: a) phase pressures;
b) water saturation; c) non-equilibrium pressure difference. . . . 149
8.1
Schematic representation of the experimental set-up (no to scale). 155
8.2
Schematic representation of the micro-model . . . . . . . . . . . 157
8.3
a) binary photoresist phase; b) binary nitrogen phase; c) binary
decane phase; d) composite image . . . . . . . . . . . . . . . . . 160
8.4
Primary and scanning drainage and imbibition points under equilibrium condition (full hysteresis loop). . . . . . . . . . . . . . . . 164
8.5
a) Comparison between external pressure difference (symbols with
a continuous line) and average capillary pressure (only symbols)
versus decane saturation; b) Specific interfacial area versus decane saturation; c) Specific interfacial area versus average capillary pressure. The plots are from equilibrium drainage experiments.165
8.6
a) Comparison between external pressure (symbols with a continuous line) and average capillary pressure (only symbols) versus
decane saturation; b) Specific interfacial area versus decane saturation; c) Specific interfacial area versus capillary pressure. The
plots are from equilibrium imbibition process experiments. . . . . 167
xvi
LIST OF FIGURES
8.7
a) Pc − S − awn surface fitted to normalized data points for equilibrium drainage experiments; b) Pc − S − awn surface fitted to
normalized data points for equilibrium imbibition experiments. .
8.8 Histogram I represents the ratio between measured specific interfacial awn throughout drainage to the interpolated values on
the imbibition surface. Histogram II represents the ratio between
the measured specific interfacial area under imbibition to the interpolated values on the drainage surface. Both are related to a
equilibrium experiments. . . . . . . . . . . . . . . . . . . . . . .
8.9 a) Pc − S − awn surface fitted to normalized data points for nonequilibrium drainage experiments; b) Pc − S − awn surface fitted
to normalized data points for non-equilibrium imbibition experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.10 Histogram III represents the ratio between measured specific interfacial awn under drainage to the interpolated values on the
imbibition surface. Histogram IV represents the ratio between
the measured specific interfacial area under imbibition to the interpolated values on the drainage surface. . . . . . . . . . . . . .
8.11 Histogram V represents the ratio between measured specific interfacial awn during nonequilibrium drainage to the interpolated
values on the equilibrium drainage surface. Histogram VI represents the ratio between the measured specific interfacial area under nonequilibrium imbibition to the interpolated values on the
equilibrium imbibition surface. Both are related to a nonequilibrium experiments. . . . . . . . . . . . . . . . . . . . . . . . . . .
8.12 Histogram VII represents the ratio between measured specific interfacial awn during nonequilibrium drainage to the interpolated
values on the equilibrium imbibition surface. Histogram VIII represents the ratio between the measured specific interfacial area under nonequilibrium imbibition to the interpolated values on the
equilibrium drainage surface. . . . . . . . . . . . . . . . . . . . .
168
170
173
174
175
176
List of Tables
3.1
3.2
Brooks-Corey parameters and density ratio. . . . . . . . . . . . .
Average values for several variables in the two grid blocks shown
in Figure 3.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
41
4.1
4.2
4.3
Fluids properties at 20◦ C. . . . . . . . . . . . . . . . . . . . . . 48
Absolute maximum pressure error of the pressure transducer in Pa. 54
TDR’s parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1
Van Genuchten parameters obtained for primary drainage, main
drainage and main imbibition. . . . . . . . . . . . . . . . . . . . .
6.1
6.2
6.3
6.4
6.5
6.6
7.1
The non-equilibrium coefficient τ estimated at the local scale versus local water saturation Sw . . . . . . . . . . . . . . . . . . . . .
Values of the non-equilibrium coefficient τ at the local scale for
different water saturations. Regression curve not forced through
w
origin, (Pn − Pw ) − Pc = −τ ∂S
∂t + I. . . . . . . . . . . . . . . . .
Van Genuchten parameters for average capillary pressure. . . . .
Non-equilibrium coefficient [τ ] at the column scale estimated at
various average water saturation < Sw > . . . . . . . . . . . . . .
Non-equilibrium coefficient < τ >i at the column scale estimated
at various average water saturation < Sw >. . . . . . . . . . . . .
Dynamic number calculate at the local and column scale. . . . .
73
107
109
111
114
114
119
Fluids and medium properties at 20◦ C, and Brooks-Corey parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
xvii
xviii
7.2
7.3
7.4
8.1
8.2
8.3
8.4
LIST OF TABLES
Intrinsic permeability k, and porosity φ of four different zones
the column between measurement levels z1 , z2 , z3 . . . . . . .
Properties of the top and bottom virtual boundary layers. . .
The non-equilibrium coefficient τ0 . . . . . . . . . . . . . . . .
in
. . 128
. . 128
. . 132
Physical properties of decane at 24.5◦ C. . . . . . . . . . . . . . .
Drainage and imbibition experiments under equilibrium condition.
Mode and average ratio between measured and interpolated specific interfacial area. . . . . . . . . . . . . . . . . . . . . . . . . .
Non-equilibrium experiments. . . . . . . . . . . . . . . . . . . . .
155
169
171
171
Chapter
1
Introduction
1.1
Motivations
Interest in understanding and predicting multiphase processes has increased immensely in the last decades due to the increase of many subsurface hydrosystem
or industrial application. Two phase flow is encountered in many artificial and
natural porous media system. For example, two-phase system are encountered
in the saturated zone when dense non-aqueous phase fluids infiltrate into the
subsurface. Dense non-aqueous phase liquids (DNAPLs) are among the most
common groundwater contaminants. They are a class of hazardous, synthetic
organic compounds that have proven to be particularly challenging to remove.
DNAPLs can reach considerable depths below the water table due to their high
density. Due to their low solubility, once in the subsurface, DNAPLs constitute
a long-term source of contamination. It has been shown that flow of immiscible fluids (e.g. DNAPL) in aquifers or in reservoirs is related to the complex
structure of the subsurface such as the presence of heterogeneities and small
lenses. Ideally a numerical model should include all these details. Commonly,
due to the limitation on characterizing the subsurface or due to the limitation
in the computational power, these small scale heterogeneities are not taken into
account. The effect that these small features have on the flow of immiscible
fluids, however, should be taken into account to describe properly the complete
processes in the subsurface. Thus, average quantities that include the effect of
these small structures are needed. This requires a proper definition of average
values for various properties. The necessity of identifying average properties is
not limited only to aquifers or reservoirs but also, for example, in upscaling from
1
2
Chapter 1. Introduction
the pore to core scale. One of the most important variables is fluid pressure.
Although at first one may think that defining an average fluid pressure is an
easy task, the answer to a simple question such as ’what is the correct average
pressure’ is not trivial.
Modeling flow and transport in a multiphase system involves a number of
constitutive relationships specific for a particular porous media and set of fluids. These relationships are of critical importance to hydrologic modeling. It
is through the incorporation of constitutive relationships into the conservation
equations that a mass balance statement is transformed into a formulation specific to the particular porous medium and fluid. In two-phase flow problems, the
flow equations for each fluid phase are written in terms of fluid phase pressures
and saturations. Traditional models suggest that under equilibrium conditions,
the pressure difference between two fluids is equal to the capillary pressure,
Pn − Pw = Pc (Sw ). This equation is adopted as a closure equation to couple the
flow equations of the two fluids. Many researchers have questioned the validity of
this relationship. In fact, it may be valid under equilibrium conditions, however,
it is used to describe processes that occur under non-equilibrium conditions.
Theory based on a thermodynamic approach has proposed an alternative closure equation which accounts for the non-equilibrium effect (Hassanizadeh and
Gray, 1990). This alternative states that the pressure difference between two
fluids under non-equilibrium conditions is related to the capillary pressure plus
an additional term which accounts for the speed at which the process occurs,
this is
∂Sw
Pn − Pw = Pc − τ
(1.1)
∂t
where Pn and Pw are the pressures of the two fluids, Pc is the capillary pressure,
w
τ is a nonequilibrium coefficient, and ∂S
∂t is the rate change of saturation.
1.2
Research Objectives
During the past several decades, several averaging operators have been introduced to obtain upscaled pressure quantities. However, the answer to a simple
question such as ’what is the correct pressure average’ is not trivial. Traditionally the intrinsic phase averaging operator is employed to obtain the macroscale
pressure. Recent theoretical studies suggested that this operator may lead to
an artifact on the average phase pressure even for the case of single-phase flow.
Other averaging operators have been also suggested. In this thesis, the effect of
various averaging operators on the average pressure and capillary pressure are
investigated.
1.3. Thesis Outline
3
One of the most important parameters in two-phase flow is the capillary
pressure, Pc . Capillary pressure is well defined at the microscale. It is a function
of the interfacial tension between two fluids and it is related to the radii of
curvature of the meniscus formed at their interface. Under equilibrium conditions
it is defined to be equal to the difference between the nonwetting and wetting
phase pressures, pc = pn − pw . For analogy with the microscale, the same
definition of capillary pressure is then adopted also at the macroscale. However,
this is not a proper definition of capillary pressure and may be valid only under
certain conditions. Experimentally, a common practice that is used to obtain
average capillary pressure is to subtract the wetting phase pressure from the
nonwetting phase pressure measured in the external fluid reservoirs. This is then
assumed to be representative value for the fluids-porous media system. However,
it is shown that this method leads to inaccuracies due to gravity effects. Thus the
question arises: ’how can we obtain averaged or an upscaled capillary pressure’ ?
The coefficient τ in Equation 1.1 relates to the strength of nonequilibrium
effect in a given system of fluids-porous medium. This coefficient τ needs to
be estimated and its functional dependence on the water saturation and on the
scale length has to be determined. This is investigated in this study using both
experimental and numerical approaches.
At the microscale, capillary pressure is defined by Young-Laplace equation
and, under equilibrium condition, is equal to the pressure difference between
two-phase fluids at the meniscus. It has been reported in a large number of experimental works that capillary pressure depends on saturation and that this relationship exhibits an hysteretic behavior. Based on a thermodynamic approach,
researchers have suggested an alternative description to the capillary pressuresaturation relationship. In that approach, the fluid-fluid interfacial area is introduced as an important state variable needed for the determining the fluid phases
distribution. According to this theory, the capillary pressure is a function of not
only saturation but also of the specific interfacial area Pc = Pc (Sw , awn ). This
approach allows for modelling the hysteretic behavior of the capillary pressure
versus saturation in an elegant way. This relationship needs to be investigated.
1.3
Thesis Outline
This thesis is organized as follows. In Chapter 2, the underlying theories and
concepts describing the flow and transport of two immiscible fluids in a porous
medium are presented. In Chapter 3, the concept of average pressure and
saturation are introduced. While averaging phase saturation is quite straightforward, the correct definition of average phase pressure is not well established
4
Chapter 1. Introduction
yet. In this chapter a simple situation is considered related to the displacement
of a nonwetting fluid by a wetting under equilibrium conditions. We know that
under equilibrium conditions, both local-scale and average phase potentials must
be constant. We calculate average fluid pressures obtained by various averaging operators and determine whether they satisfy the constant phase potential
criterium.
A series of laboratory experiments were carried out at the column scale in
order to investigate equilibrium and non-equilibrium effects on two-phase flow.
In Chapter 4, the experimental set-up, including instrumental device and experimental procedure are presented.
In Chapter 5, results of primary drainage, main drainage and main imbibition experiments under equilibrium conditions are presented. These include
local-scale phase pressures and saturation. The corresponding column-scale values are obtained by averaging them. The results of the various averaging operators are discussed.
In Chapter 6, results from nonequilibrium two-phase flow experiments are
presented. These regard primary drainage processes at different injection pressures. The local phase pressures and saturations are then averaged to obtain
corresponding column-scale values. The values of the non-equilibrium coefficient τ and its functional relationship to saturation are determined at both local
and column scales.
Beside laboratory work, numerical simulations were carried out to simulate
nonequilibrium two-phase flow experiments. The traditional theory was compared with a new approach which accounts for the nonequilibrium effects. In
Chapter 7, results from the two types of model formulations are compared with
the experimental data and discussed.
Theoretical work suggested that the hysteretic behavior observed in the
capillary-pressure-saturation relationship is a artifact of the projection of three
dimensional Pc − Sw − awn surface on the capillary pressure-saturation plane.
Previous experiments had shown that this conjecture holds, within measurements errors under equilibrium conditions. However, whether this is also the
case under nonequilibrium conditions has not been shown. Thus, the question
arises whether the capillary pressure-saturation relationship is the same for equilibrium and non-equilibrium processes. In Chapter 8, the capillary pressure saturation - specific interfacial area relationship is investigated experimentally in
a two-dimensional micro-model. The experimental set-up and methodology are
described and results are presented. Finally in Chapter 9, a summary of the
findings from the numerical and experimental analysis are given and conclusions
are drawn.
Chapter
2
General theory of two-phase
flow
The interest in understanding and properly describing multiphase flow in porous
media has increased immensely over the last decades due to the importance of
many engineering applications such as remediation of polluted sites, paper production, fuel cells, enhance oil recovery, etc. Various theoretical, numerical, and
experimental studies have been conducted to properly describe these complicated
processes. Many of these investigations have contributed to understanding several aspects related to the intricate processes of multiphase flow and transport
in porous media system. In this chapter, an overview of the fundamentals related to the physical processes that underlie the flow of immiscible fluids at the
micro-scale and macro-scale is given. Obviously, it is not possible to include all
details of such processes. Therefore, the intention is to present the underlying
theories and concepts relevant in the context of this thesis.
2.1
Interfacial tension and wettability
Two important phenomena occur when two immiscible fluids are in contact
with each other in a porous medium. These are known as interfacial tension
and wettability. Interfacial tension is caused by the attraction between the fluids
molecules by various intermolecular forces. In the bulk of each fluid, cohesion
forces work between the molecules of the fluid. These can be seen as the tendency
of the material to hold itself together. Cohesive forces act uniformly within a
liquid phase resulting in a zero balance of forces. In addition, there are inter5
6
Chapter 2. General theory of two-phase flow
molecular forces between molecules of the two fluids called adhesion (Jensen and
Falta, 2005). Thus, at the surface between two fluids, the molecules are subject
to both cohesive and adhesive forces. The cohesive forces within the fluid are
much stronger than the adhesive force between the fluids. As consequence there
is an imbalance of forces at the interface. To reduce the forces, a fluid phase aims
at minimizing its surface area. The interfacial tension γ, between phases a and
b relates the work required to minimize the surface to change area Aab between
the phases. This is a force tangent to the interface separating the two phases
(force per unit length). The interfacial tension is specific for a pair of substances
and depends on the chemical composition of the phases and the temperature.
Two immiscible fluids in contact with each other in a porous medium compete
against each other for contacting the surface of the solid grains. The wettability
is the affinity that a phase exhibits for the grains. The wettable phase will tend
to coat the grain while the other phase is displaced toward the middle of the
pore space. This is caused by the attractive forces between different types of
molecules. Thus, by the term wetting phase, we identify a phase which exhibits
a strong attraction towards the solid phase. In contrast, by the term nonwetting
phase, we define the phase which manifest least preference towards contact with
the grains or solid phase. The order of wettability of phases has a large influence
on their distribution and thus on their mobility and retention characteristics in
a porous medium. The interface between two fluids forms a contact angle θ with
the solid surface. A contact angle θ < 90 indicates that a fluid is the wetting
phase.
2.2
Concept of capillary pressure at the microscale
In a two-fluids-solid system, three interfaces are formed: between the two fluids
and between each fluid and the solid phase. The curve where all of three interfaces intersect each other is called common line, see Figure 2.1. The common
line can be seen as a transition zone where the molecules of the two fluids and
the solid interact. In this system, three interfacial tensions are distinguished: between the wetting and the nonwetting phases, γwn , between the wetting phase
and the solid phase, γws , and between the nonwetting and the solid phase, γns .
The movement of the fluid-fluid interface is governed by the balance of forces
exerted by the two fluids on the interface and the forces present within the interface. The interface between the fluid phases forms a meniscus with a certain
curvature. Associate with a meniscus and its curvature, there exists the capillary
pressure pc defined by as follows:
2.2. Concept of capillary pressure at the microscale
7
Figure 2.1: Two immiscible fluids is a capillary tube.
pc = γwn
1
1
+
rx
ry
=
2γwn
Rm
(2.1)
where rx and ry denote the principal radii of curvature of the meniscus, and
RM is the mean radius of curvature. Thus, through rx and ry the capillary
pressure depends on the pore dimension and through the interfacial tension it
depends on the surface properties of the fluids and the soil. Because γwn relates
to the change of interfacial free energy per unit change in interfacial area, pc as
defined in Equation 2.1 may be viewed as a state variable of the interface. Thus,
the capillary pressure as expressed in Equation 2.1 is an intrinsic property of
the two fluids and the solid phase and it is valid whether the interface is moving
or not. The pressures of the fluids on the either side of the interface are not
equal. The pressure difference under equilibrium conditions is found to be equal
to capillary pressure pc :
pc = pn − pw
(2.2)
where pn and pw represent the pressure of each fluid on the side of the meniscus.
Note that pn − pw is not a state variable of the interface. In fact the relationship
between pn − pw and pc is not an intrinsic property of the medium and the pair
of fluids; it depends on external factor and flow conditions. This relationship
can be derived from the force balance in the direction normal to the interface. In
this respect, Hassanizadeh and Gray (1993) derived the following force balance
8
Chapter 2. General theory of two-phase flow
equation for the direction N normal to an interface:
pn − pw =
2γwn
+ N · (τn − τw ) · N − ∇σ · τwn · N
Rm
(2.3)
where τn and τw are viscous stress tensors of the n-phase and w-phase on the
two sides of the meniscus, and τwn denotes the interfacial viscous force present
when the interface is in motion. Even if interfacial viscous forces are negligible
or not present at all, pn − pw may be different form pc due to dissipative effects
in the fluids. In such a case, pn − pw will not be equal to pc . This has been
shown by Sheng and Zhou (1992), who simulated the motion of a meniscus in a
tube during piston-displacement of a wetting phase by a nonwetting phase and
vice versa. They found
pn − pw =
µq A
2γwn
±B
Rm
r
(2.4)
where pn and pw are the average value on the two sides of the interface, µ is viscosity, q is velocity, B and A are coefficients that control the velocity-dependent
’capillary pressure’. Thus the relationship pn − pw = pc is valid at the meniscus
only under static condition. Under dynamic conditions, pn − pw depends on the
flow velocity, which at larger scales manifest itself as a change of saturation with
time.
2.3
Capillary pressure at the macroscale
The question arises: what is actually the capillary pressure at the macroscale?
Many attempts have been made in the last few decades to define the capillary pressure at the macroscale. Bear and Bachmat (1986) , Bear and Verruijt
(1987), by analogy with the microscale capillary pressure, defined the macroscopic capillary pressure as the difference in pressures of the nonwetting and
wetting phases:
< Pc >=< Pn > − < Pw >
(2.5)
where Pc denotes the macroscopic capillary pressure, and < Pn > and < Pn >
are the average phase pressures. Thus, the pressure of the wetting and nonwetting phases are averaged over a portion of volume where each phase is actually
present. However, Equation 2.1 applies at the microscale and it regards only
2.3. Capillary pressure at the macroscale
9
the interface. Thus Pc could be averaged only over an interface but not over
a volume. Many models have been proposed that relate the capillary pressure
to saturation, Pc = f (Sw ). These models, however, are not physically based
but are empirical relationships. Thus, they do not contain, explicitly, all the
effects and processes that influence the distribution of immiscible fluids in a
porous medium (e.g. immiscibility, surface tension, fluid-fluid interface). Commonly Pc −Sw curves presented in the literature show that at irreducible wetting
saturation Pc goes to infinity. The capillary pressure is commonly determined
indirectly by measuring the pressures of the two fluids in external reservoirs bordering a porous medium. Numerous comments in the literature indicate that
externally measured capillary pressure near residual saturation loses significance
with respect to the condition within the porous medium (Harris and Morrow ,
1964; Morrow and Harris, 1965). For example, the wetting phase near irreducible saturation becomes disconnected and thus loses its hydraulic connection
with the external reservoir. The disconnected wetting phase will be completely
surrounded by the non-wetting phase. This means that an increase in the nonwetting phase pressure causes an increase in the wetting phase pressure and thus
it does not result in an appreciable increase in the capillary pressure within the
porous medium. Several empirical relationships between capillary pressure and
the phase saturation were proposed. The most widely used model are those
suggested by Brooks and Corey (1964) and Van Genuchten (1980). Brooks and
Corey (1964, 1966), based on the comparison of a large number of experimental
data, suggested the follow relationship:
−1
Pc (Se ) = Pd · Se λ
(2.6)
where Pd denotes the entry pressure, identified as the pressure beyond which
the nonwetting phase first infiltrates inside the pores of the medium. The coefficient λ indicates the pore size distribution index. High values of the coefficient
is interpreted as indicative of a narrow distribution of the pore size. Se is the
effective saturation, defined as:
Se =
Sw − Swr
1 − Swr − Snr
Swr ≤ Sw ≤ 1
(2.7)
where Swr is the irreducible wetting phase saturation and Snr is the residual
nonwetting phase saturation. Another model was suggested by Van Genuchten
(1980):
10
Chapter 2. General theory of two-phase flow
Pc (Se ) =
1 −1/m
1
(S
− 1) n
α e
(2.8)
where α [1/Pa] can be interpreted as the inverse of the entry pressure. The
parameters n and m can be related to the pore size distribution and are related
to the relationship by m = 1 − 1/n. Another formula commonly used to define
capillary pressure at the macroscale is due to Leverett (1941):
Pc = γwn
ǫ 12
k
J(Sw )
(2.9)
where k is the permeability of the medium and J(Sw ) is called J-Leverett function and it is a dimensionless function of saturation and is independent of the
soil properties. This expression is also empirical in nature. Other more physically based definitions of the capillary pressure were given by Kalaydjian (1987)
and Pavone (1989). In their approach, bulk phases, interfaces, and common
lines were modeled and constitutive relationships describing the behavior of the
system were developed at the macroscale. A shortcoming of their approach was
that fluid pressure was introduced at the microscale and therefore a number
of thermodynamic relationship known for a single phase fluid continuum were
assumed to be valid also for a multiphase medium at the macroscale. Hassanizadeh and Gray (1990) based on thermodynamic approach, derived a physically based macroscale capillary pressure relationship. In their approach, contrary to Kalaydjian (1987) the pressures of the phases were introduced directly
at the macroscale by a volume averaging procedure. A more detailed description
of their approach is given further in this chapter.
2.4
Hysteresis
An important feature of the macroscopic capillary pressure-saturation curve is
its hysteretic behavior observed when reversing the flow direction (e.g. from
drainage to imbibition). Many authors agreed that the cause of hysteretic behavior at the macroscale has to be searched already at the pore scale. At the
microscale, many authors attributed this hysteretic behavior to the nonuniformity of the pores, the hysteresis in contact angle, entrapped phases, and ’Haines
Jumps’ (Bear , 1972, 1979; Corey, 1977). The hysteresis observed in the raising
and lowering of the meniscus in a vertical capillary tube is commonly called
contact angle hysteresis. Many researchers (Hillel , 1980; Miller and Noegi,
1985; Schiegg, 1986) attributed the contact angle hysteresis to the roughness
2.5. The relative permeability-saturation relationship
11
of the solid surface, adsorption effects, and surface impurities. (Hassanizadeh
and Gray, 1993), considered the movement of an interface in a capillary tube.
They suggested that the contact angle hysteresis was due to the fact that the
stresses in the solid-fluid interfaces would oppose the movement of the contact
line (and translation of the interface). Thus, the existence of advancing and
receding contact angles could not be explained without taking fluid-solid interfacial elastic stresses into account. In other words, the hysteretic behavior of the
capillary pressure is related to the configuration of interfaces. Based on thermodynamic derivation, the authors concluded that the hysteretic behavior of
the capillary pressure-saturation relationship can be modelled by including the
specific interfacial area in the formulation.
2.5
The relative permeability-saturation relationship
When two phases are simultaneously present in a pore, the presence of one
influences the flow behavior of the other phase. In fact, the pore space is not
available to one phase only but it is shared. This leads to the common concept
of relative permeability which is defined as the ratio of the permeability of a
phase kα at a given saturation to the intrinsic permeability k :
krα =
kα
k
0 ≤ krα ≤ 1
(2.10)
Relative permeability is a function of saturation and it relates to a specific set
of porous media and fluids. Two models are usually adopted to describe the
relationship between relative permeability and saturation: Burdine (1953) and
Mualem (1976) models. Burdine considered a model where the radii of the tubes
vary only perpendicular to the flow direction while Maluem derived the relative
permeability for a bundle of capillary tubes, that also vary in the flow direction.
The Burdine model reads as:
2+3λ
λ
krw = Se
(2.11)
and for the nonwetting phase
krn = (1 − Se )2 (1 − Se )
2+λ
λ
(2.12)
12
Chapter 2. General theory of two-phase flow
Later Van Genuchten (1980) applied his water retention curves to the model
of Maluem. He derived an analytical expression for the following relationship:
m i2
h
1
krw = Se 1 − 1 − Sem
(2.13)
and for the nonwetting phase:
2m
1
1
krn = (1 − Se ) 2 1 − Sem
2.6
2.6.1
(2.14)
Two-phase flow equations at the macroscale
Darcy’s law for two phase flow
The basic equation for fluid flow in porous media is the so called Darcy’s law
(1856)
q = −K · ∇h
(2.15)
where q is the Darcy velocity vector, K is the hydraulic conductivity tensor,
and ∇h is the hydraulic gradient. For an isotropic medium, K is a scalar and
is related to the properties of the fluid (density and viscosity) and the porous
media K = kρg/µ. Darcy’s law has been also formulated in term of pressure
instead of hydraulic gradient as follows:
k
q = − (∇P − ρg)
µ
(2.16)
where k denotes the intrinsic permeability, µ is the fluid’s viscosity, P is the
fluid pressure, and g is the gravity acceleration vector. This equation was meant
to describe the flow of water in one direction in a homogeneous porous medium.
This was then extended to a more complex system such as for example, the
simultaneous flow of two or more immiscible fluids. The generalized form of
Darcy’s law can be expressed in the following form:
qα = −
krα k
(∇Pα − ρα g)
µα
(2.17)
This extension of Darcy’s law is based on the assumption that the driving force
for the α phase is only the gradient in the phase pressure and the gravitational
force, and that the relative permeability is a function of saturation.
2.7. Two-phase flow theory after Hassanizadeh and Gray (1990)
2.6.2
13
Mass conservation
The mass balance equation for α phase is given as:
∂Sα φρα
+ ∇ · (ρα qα ) = 0
(2.18)
∂t
Assuming that the porous medium and the fluid phase are incompressible and
substituting Equation 2.17 in the mass balance Equation 2.18 for each phase
yield the flow equations for the wetting and non-wetting phases:
∂Sw
krw k
φ
−∇·
(∇Pw − ρw g) = 0
∂t
µw
(2.19)
∂Sn
krn k
−∇·
(∇Pn − ρn g) = 0
∂t
µn
(2.20)
and
φ
Assuming density, viscosity, porosity, intrinsic permeability and relative permeability are known, there are four unknowns in Equations 2.19 and 2.20 Pn ,
Pw , Sn , Sw . To solve this set of coupled partial differential equations, two additional equations are required. One equation is obtained from the total volume
balance for the phases
Sw + Sn = 1
(2.21)
The second equation is the traditional approach of defining the capillary pressure
as the pressure difference between two phases
Pn − Pw = Pc (Sw )
(2.22)
Based on Equations 2.21 and 2.22, the number of unknown variables are reduced to two.
2.7
Two-phase flow theory after Hassanizadeh
and Gray (1990)
Researchers agree that, to obtain a physically well-founded theory that can explain all the observed phenomena, the thermodynamics and the geometry of the
14
Chapter 2. General theory of two-phase flow
interfaces between the phases must be considered. Obviously, at the macroscale,
single interfaces are not identifiable. Nevertheless, their effect should be incorporated into the full macroscopic description of multiphase flow. Hassanizadeh
and Gray (1990), developed a macroscopic thermodynamic theory to describe
multiphase flow in porous media. Basic principles of mass, momentum, and
energy conservation were used to obtain equations that account for all bulk
phases, interfaces and contact lines. The hypothesis made in their theory to
derive constitutive relationship is the dependence of the Helmholtz free energy
of each phase and interface on the mass density ρα , temperature T, saturation
Sα , porosity φ, specific interfacial area aαβ , and the solid phase strain tensor Es
(Hassanizadeh and Gray, 1990). It reads as :
An
= An (ρn , T, awn , ans , Sw , φ)
(2.23)
Aw
= Aw (ρw , T, awn , aws , Sw , φ)
(2.24)
As
= As (ρs , T, ans , aws , Es , φ)
(2.25)
Aαβ
= Aαβ (Γαβ , T, φ, aαβ , Sw )
with
αβ = wn, ws, ns (2.26)
The inclusion of the interfacial area is known to have an important role on the
thermodynamic behavior of the system. It is reported that hysteresis (in capillary pressure) was caused by instability of the interface configurations (Scheidegger , 1974), and that interfacial energy can have a large influence on the state
of the entire porous body. Based on the constitutive relations 2.23, 2.24, 2.25,
Hassanizadeh and Gray (1990), derived the following residual entropy inequality
−
DSw
∂Aw
∂An
∂Aαβ
φ(Pn − Pw ) + φSw ρw
+ φSn ρn
+ Σαβ aαβ Γαβ
≥0
Dt
∂Sw
∂Sw
∂Sw
(2.27)
w
where DS
Dt is the material time derivative of the wetting phase saturation, and
Pn and Pw are the macroscopic pressure of the n-phase and w -phase, respectively. The macroscopic pressures were defined at the macroscale based on the
thermodynamic theory and were not based on any averaging procedure from the
microscale (Hassanizadeh and Gray, 1990). Then they defined:
∂Aw
∂An X aαβ ραβ
Pc = −Sw ρw
− S n ρn
−
∂Sw
∂Sw
φ
αβ
∂Aαβ
∂Sw
(2.28)
2.7. Two-phase flow theory after Hassanizadeh and Gray (1990)
15
Thus, the macroscopic capillary pressure is related to the change in free energy
of phases and interfaces as a result of change in the saturation. Based on this
approach the macroscopic capillary pressure is found to be an intrinsic property
of the fluids and the porous medium, depending on the following quantities
Pc = f (ρn , ρw , T, Γwn , Γws , Γns , awn , ans , aws , φ, Sw )
(2.29)
This relationship allows for a physical interpretation of the capillary pressure
instead of simply identifying it as the difference of pressures between two fluids.
The dependence on ρα , Γαβ , φ indicates that the capillary pressure-saturation
curves are specific to a particular set of fluid and solids, are dependent on the interface composition, and may vary when the porosity changes when the medium
is deformable. The dependence of Pc on aαβ is a macroscale manifestation of
the dependence of the capillary pressure on the curvature of interfaces at a given
saturation. Equation 2.29 shows that the capillary pressure is a function of
eleven independent variables. Simple case can be considered; e.g. when the fluid
densities, mass densities of interfaces, temperature and porosity are constant.
Assuming that the the porous medium is coated by the wetting phase, then the
specific interface ans = 0 and the aws is constant. Thus, after these consideration
the capillary pressure can be expressed as:
Pc = f (awn , Sw )
(2.30)
Thus the capillary pressure is recognized as a function of two independent variables, saturation and fluid-fluid specific interfacial area. The authors suggested
that the hysteretic capillary pressure-saturation behavior can be interpreted as
the projection of the Pc − Sw − awn surface onto the Pc − Sw plane. Substituting
Equation 2.28 into Equation 2.27 the entropy inequality can be written as:
−
∂Sw
[(Pn − Pw ) − Pc ] ≥ 0
∂t
(2.31)
If Pn − Pw is greater than the capillary pressure at a given saturation, then
w
the ∂S
∂t must be negative. In this case drainage occurs. Similarly imbibition
occurs when the Pn − Pw is smaller than the capillary pressure at a given satw
uration, then ∂S
∂t must be positive, i.e imbibition should occur. Equation 2.31
suggests the following linear approximation:
16
Chapter 2. General theory of two-phase flow
−
∂Sw
1
= − [(Pn − Pw ) − Pc ]
∂t
τ
(2.32)
where τ is a nonnegative material coefficient. The coefficient τ may be interpreted as a measure of the speed at which a change in fluid pressure causes a
change in fluid distribution. If τ is found to be small, the equilibrium between
Pn − Pw and Pc will be reestablished quickly after pressure are change. This
coefficient may be a function of ρα , φ, Sw , aαβ , and Γαβ .
BIBLIOGRAPHY
17
Bibliography
Bear, J. (1972), Dynamics of Fluids in Porous Media, Dover Publ.
Bear, J. (1979), Hydraulics of groundwater flow, New York.
Bear, J., and Y. Bachmat (1986), Macroscopic modelling of transport in porous
media, pt. 2: Applications to mass, momentum and energy transport, Transport in Porous Media, 1 (3), 241–270.
Bear, J., and A. Verruijt (1987), Modeling groundwater flow and pollution,
D.Reidel, Norwell, Mass.
Brooks, R., and A. Corey (1964), Properties of porous media, In Hydrol. Pap.,
volume band 3 (Colorado State University, Fort Collins.).
Brooks, R., and A. Corey (1966), Properties of porous media affecting fluid flow.,
Journal of the irrigation and drainage division.
Burdine, N. (1953), Relative permeability calculations from pore size distribution data, Transaction of the American Institute of Mining and Metallurgical
Engineers, 198.
Corey, A. (1977), Mechanics of heterogeneous fluids in porous media, Water
Resources Pubblications, Fort Collins, Colo.
Harris, C., and N. Morrow (1964), Pendular moisture in packing of equal spheres,
Nature, 203.
Hassanizadeh, S., and W. Gray (1990), Mechanics and thermodynamics of multiphase flow in porous media, Advances Water Resources, 13.
Hassanizadeh, S., and W. Gray (1993), Thermodinamic basis of capillary pressurein porous media., Water Resources Research., 29.
Hillel, D. (1980), Fundamentals of soil physics, Academic San Diego Calif.
Jensen, K., and R. Falta (2005), Soil and Groundwater Contamination: Nonaqueous Phase Liquids, AGU Books Board.
Kalaydjian, F. (1987), A macroscopic description of multiphase flow involving
spacetime evolution of fluid/fluid interface, Transport Porous Media, 2.
Leverett, M. (1941), Capillary behavior in porous media, Trans.Am.Inst. Min.
Metall. Pet. Eng., 142.
18
Chapter 2. General theory of two-phase flow
Miller, C., and P. Noegi (1985), Interfacial phenomena, Marcel Deker New York.
Morrow, N., and C. Harris (1965), Capillary equilibrium in porous media, Soc.
Petl. Eng. J., 5.
Mualem, Y. (1976), A new model for predicting the hydraulic conductivity of
unsaturated porous media., Water Res. Res., 12 (3), 513–522.
Pavone, D. (1989), Macroscopic equations derived from space averaging for immiscible two-phase flow in porous media, Rev.Inst.Fr.Pet., 44 (1), 29–41.
Scheidegger, A. E. (1974), The physics of flow through porous media, University
of Toronto Press, Toronto, Ont.
Schiegg, H. (1986), Evaluation and treatment of cases of oil damage with regard
to groundwater protection, Albertson et al., Rep. LTWS 20, Swiss federal
Office of the Environment, Zurich.
Sheng, P., and M. Zhou (1992), Immiscible-fluid displacement: Contact-line
dynamics and the velocity-dependent capillary pressure, Physical Review A,
45 (8), 5694–5708.
Van Genuchten, M. (1980), A closed-form equation for predicting the hydraulic
conductivity of unsaturated soils, Soil Sci.Soc. Am. J., 44.
Chapter
3
What is the correct
definition of average
pressure?1
Use of a correct definition of average pressure is important in numerical modelling of oil reservoirs and aquifers where the simulated domain can be very large.
Also, the average pressure needs to be defined in the application of pore-network
modelling of (two-phase) flow in porous media, as well as the (theoretical) upscaling of flow equations. Often the so-called intrinsic phase-volume average
operator is employed. Here, we introduce and investigate several alternative averaging operators. We consider static equilibrium of two immiscible fluids in a
homogeneous, one-dimensional, vertical porous medium domain under a series
of (static) drainage conditions. It is argued that the correct average pressure
must preserve the fact that fluid potentials are constant. It is found that the
centroid-corrected phase average pressure fulfils this requirement. However, early
in the drainage process, when the nonwetting phase front has not infiltrated very
far into the domain, this averaging operator can give rise to negative average
capillary pressures. This occurs especially when the domain and/or density differences are large, or when the entry pressure is small. The negative average
capillary pressures occur because the nonwetting phase pressure effectively is
projected towards the centroid of the domain.
1 Accepted for publication in Transport in Porous Media as: S. Korteland, S. Bottero, S.M.
Hassanizadeh, C.W.J. Berentsen, What is the correct definition of average pressure?
19
20
3.1
Chapter 3. What is the correct definition of average pressure?
Introduction
When numerical models are used to investigate flow processes in reservoirs or
aquifers, the simulated domain commonly has a size on the scale of hundreds
of meters to kilometers. However, oil reservoirs and aquifers usually have complex structures, consisting of heterogeneities on different length scales. In an
ideal situation, the numerical flow simulator should include all these variations.
Unfortunately, the current computational power and our ability to characterize
the subsurface are not sufficient to include such detailed structures. As a result,
the numerical grid size is often much larger than the scale of medium heterogeneities. For instance, for a medium-sized oil reservoir, the grid in a numerical
simulator may have a resolution on the decameter scale, whereas heterogeneities
may exist at the scale of centimeters. For such grid blocks, average parameter
values are needed. These upscaled parameters should ideally be derived from
the fine-scale parameters inside the grid blocks. This requires appropriate definitions of average values for various properties. Also, in pore-network modelling
of (two-phase) flow in porous media, average values need to be specified. Moreover, in theoretical upscaling of (two-phase) flow, starting from Navier-Stokes
equation or Darcy-flow equation, average properties need to be defined. One of
the most important variables is fluid pressure. We submit that it is not clear yet
what the correct definition of average pressure should be. Recently, there has
been some discussion in literature in this regard (Gray and Miller , 2004, 2007;
Nordbotten et al., 2007, 2008).
In the vast literature on porous medium averaging from the pore scale to
the Darcy scale, almost invariably the intrinsic phase-volume average pressure is
employed for defining the macroscale pressure (see Whitaker , 1977; Gray, 1975;
Gray and O’Neill , 1976; Neumann, 1977; Quintard and Whitaker , 1994,b; Bear
and Bachmat , 1986; Bear , 1972; Bachmat and Bear , 1986; Gray and Miller ,
2007)
< Pα >i =
1
Vα
Z
Pα dV
(3.1)
Vα
where Pα is the pressure of phase α, and Vα denotes the domain (and the volume) occupied by the α-phase. It is almost always (implicitly) suggested that
the intrinsic phase-volume average pressure corresponds to the pressure that we
actually measure (see Gray and Miller , 2007). To our knowledge, however, this
presumption has never been proven by means of measurements. Ideally, such
an experiment would constitute measuring pressure inside pores of a given volume by a small-scale sensor and measuring the macroscale pressure of the whole
3.1. Introduction
21
volume by a large scale sensor. Although such an experiment has not been carried out yet, we can identify realistic situations where pressure obtained from
a certain way of measurement will not be equal to the intrinsic phase-volume
average pressure. Consider the simple case of hydrostatic equilibrium in a satu-
Figure 3.1: a) schematic representation of manometer used to measure pressure
at a given elevation z1 in a porous medium column fully saturated with water
(drawing is not to scale); b) schematic representation of pore distribution at the
manometer opening, showing larger pores in the lower half of the opening; c)
pressure distribution across the manometer opening.
rated soil column as depicted in Figure 3.1a. The pressure at a given elevation
z1 can be measured by a manometer. The pressure at this elevation is equal
to ρw gh1 where ρw is mass density of water. Obviously, this is the average
pressure over the opening of the manometer, centered at z1. It is also equal to
the point pressure at the centroid of the opening, z1 (see Figure 3.1b). Now,
assume that a closer examination of the porous medium shows that the pore
sizes are not uniformly distributed over the manometer opening area, but, as
shown schematically in Figure 3.1c, there are larger pores in the lower half of
22
Chapter 3. What is the correct definition of average pressure?
the opening. This means that, because of the hydrostatic gradient in pressure,
the intrinsic phase-volume average pressure will be larger than ρw gh1, which is
the pressure measured by the manometer. The reason for this discrepancy is that
the centroid of the water phase (in this case the pore space) does not coincide
with the centroid of the averaging volume (i.e. the manometer opening). In fact,
the intrinsic phase-volume average pressure will be equal to the point pressure
at zc, the centroid of the pore space. A similar discussion on this issue can be
found in Nordbotten et al. (2007), who averaged the microscale flow equation for
cases with a porosity gradient. They showed that the classical definition of average pressure (the intrinsic phase-volume average) does not lead to the classical
Darcy’s law, but gives rise to a nonphysical gravity term. Due to the gradient in porosity, there is a systematic length-scale-dependent difference between
the centroid of the phase and the centroid of the averaging volume. Nordbotten
et al. (2007) proposed a new definition for average pressure, which was based
on a Taylor expansion around the intrinsic phase-volume average of the microscopic pressure. Gray and Miller (2004) also considered single-phase flow in a
domain with a macroscale gradient in porosity. They average Darcy-scale flow
equations and showed that intrinsic phase-volume average pressure incorrectly
prescribes a flow that does not actually exist. Their conclusion was that one has
to work with average hydraulic head (pressure head + gravity head) instead of
average pressure. But, this is not feasible in the case of two-phase flow because
we need to work with fluid pressures in order to be able to define capillary pressure. Now, consider averaging two-phase flow problems. Even when the pore
space in an averaging volume is uniformly distributed, it is usually not uniformly
occupied by the two phases. Therefore, in most cases the centroids of the two
phases do not coincide with the centroid of the averaging volume. In such cases,
it is not clear that the intrinsic phase-volume average pressure corresponds to
the measured macroscale pressure. In a second paper, Nordbotten et al. (2008)
extend their new definition of pressure to multi-phase flow in porous media. As
these authors note, even in the case of homogeneous porous media, the intrinsic
phase-volume average pressure leads to additional terms in Darcy’s law because
in general there will be gradients in saturation and pressure on all scales. The
new average defined by Nordbotten et al. (2008) is one of the averaging operators investigated in this work. Here upscaling of fluid pressure from the Darcy
scale to larger scales is investigated for a two-phase flow problem representing a
static primary drainage experiment. Five different averaging operators are used
to obtain average pressure. These operators will be introduced and discussed.
3.2. Averaging operators
3.2
23
Averaging operators
Consider a porous medium for which the porosity and capillary pressure-saturation
relationship are known. Without loss of generality it is assumed that these properties do not change with time and that porosity is constant in space. For given
boundary conditions,saturation and fluid pressure distributions under static conditions one can be derived. Thus, it is assumed that saturations Sα and pressures
Pα are known functions of time and space. Traditionally, the macroscale pressure
is defined as the so-called intrinsic phase-volume average pressure (Equation 3.1).
For two-phase flow in a porous medium with constant porosity, this averaging
operator becomes:
Z
1
i
R
< Pα > =
Sα Pα dV ;
α = w, n
(3.2)
S dV V
V α
where Sα is the saturation of phase α. Phase α is either wetting (w) or nonwetting (n). Although the integral limit indicates that the averaging is carried
out over the whole domain, in fact the averaging is performed only over that
part of the domain where the α-phase is actually present (Sα > 0). In early
stages of a front displacement, this could be only a small part of the whole
domain for the nonwetting phase. This averaging operator is analogous to the
intrinsic phase-volume average operator used in upscaling from the pore scale
to the Darcy scale (see Gray, 1975; Whitaker , 1977; Hassanizadeh and Gray,
1979). In addition, this averaging operator has been used for averaging from
Darcy scale to higher scales (Ataie-Ashtiani et al., 2001, 2002; Das et al., 2004;
Manthey et al., 2005), and in pore-network models for determining macroscale
pressure fields (Dahle and Celia, 1999; Gielen et al., 2004, 2005; Gielen, 2007).
As explained in the introduction, the average pressures defined by Equation 3.2
are actually not calculated over the same domain whenever a front exists. Also,
in general, the centroids of the two phases do not necessarily coincide with each
other or with the centroid of the sample. To alleviate partially this problem, we
may choose to define an average pressure without weighting it with the phase
volumes: Simple Phase-Average pressure express as following:
Z
1
sp
R
Pα ηα dV ;
α = w, n
(3.3)
< Pα > =
V
V ηα dV
where ηα represent an indicator function, defined as:
ηα = 1
if
Sα > 0
(3.4)
24
Chapter 3. What is the correct definition of average pressure?
ηα = 0
if
Sα < 0
(3.5)
The indicator function ηα ensures that averaging is performed over regions where
the α-phase is actually present. We shall refer to this average pressure as the
simple phase average pressure. While this average does not weight pressure by
the phase volume (i.e. saturation), it is still defined at the centroid of the of
the actual domain occupied by a phase and not at the centroid of the averaging
volume. Therefore, we consider yet another definition, for which averaging is
carried out over the whole domain:
1
V dV
< Pα >s = R
Z
Pα dV ;
α = w, n
(3.6)
V
Here it is assumed that Pα is defined everywhere in the domain of interest even
if the α-phase is not present everywhere. In the definition above, the averaging
domain is exactly the same for both the nonwetting and wetting phase. This
average pressure will be referred to in this paper as the simple average pressure.
As mentioned in the introduction, a new definition of macroscale quantities has
been proposed by Nordbotten et al. (2008). In fact, these authors introduced a
family of macroscale pressures [Pα ]n , where n refers to the order of the approximation to a smooth macroscale function [Pα ]. Nordbotten et al. (2008) obtained
an expression for macroscale pressure from its microscale equivalent. However,
their approach and definition can also be used for upscaling from the core scale to
higher scales. We propose to use the first-order approximation of the macroscale
pressure, given by:
[Pα ] =< Pα >i +
1
< z > − < zα >i · ∇ < Pα >i
∇. < zα >i
(3.7)
where < Pα >i is the intrinsic phase-volume average pressure of phase α defined by Equation 3.2, < z > is the position to which [Pα ] is assigned (usually
the centroid of the sample REV), and < zα >i is the intrinsic phase-volume
average of position vector z. Equation 5.12 suggests that the first order approximation of the macroscale pressure is equal to the intrinsic phase-volume
average pressure < Pα >i , corrected for the distance between the centroid of
the averaging volume, <z> , and the centroid of the phase, < zα >i . We refer
to this macroscale pressure as the centroid-corrected phase average pressure. In
Equation 5.12, derivatives of the intrinsic phase-average pressure < Pα >i and
3.3. Model description
25
position vector < zα >i are needed. These can be calculated for the particular
case of a vertical domain with flow from bottom to top as follows (see Appendix
9 for the derivation of these equations):
∂
1
< Pα >i =
− < Pα >i Sαtop − Sαbot ) + (Sαtop · Pαtop − Sαbot · Pαbot
∂z
< Sα > H
(3.8)
∂
1
< zα >i =
− < zα >i Sαtop − Sαbot + Sαtop · zαtop − Sαbot · zαbot
∂z
< Sα > H
(3.9)
where H denotes the length of the domain. The superscripts ’top’ and ’bot’
refer to the value of the variable at top and bottom of the domain, respectively.
Finally, also the average saturation will be used, which follows directly from its
definition of being the volume of the fluid phase divided by the volume of the
pore space. Thus:
Z
1
R
< Sα >=
ǫSα dV ;
α = w, n
(3.10)
V
V ǫdV
that assumes a constant porosity.
3.3
3.3.1
Model description
Pressure distribution across the domain
Two-phase flow problem, consisting of static primary drainage in a one-dimensional
homogeneous vertical domain is considered. Because the domain is vertical, gravity forces will influence the pressure and saturation distributions. The domain
is initially fully saturated with the water phase. Primary drainage is initiated
by forcing the nonwetting phase into the domain from below. For a given overpressure at the bottom boundary, pressure and saturation distributions can be
obtained assuming static equilibrium (i.e. no flow of both wetting and nonwetting phases). A schematic presentation of pressure distributions, for the case
where the nonwetting phase density is higher than the wetting phase density, is
shown in Figure 3.2. At static equilibrium, the wetting phase pressure distribution will be hydrostatic. Assuming zero pressure at the top of the domain, the
pressure distribution, is given by:
26
Chapter 3. What is the correct definition of average pressure?
Figure 3.2: schematic pressure distribution for the wetting and nonwetting phase
in a vertical one-dimensional homogeneous domain at static equilibrium. The
nonwetting phase has a higher density than the wetting phase, and infiltrates the
domain from below. a) The nonwetting phase has infiltrated the domain up to
zf ; b) the nonwetting phase has infiltrated the domain completely.
Pw = ρw g(H − z);
0≤z≤H
(3.11)
where g is the gravitational constant and ρw is the wetting phase density. This
expression can be made dimensionless by dividing both sides by ρw gH:
P w′ (z ′ ) = 1 − z ′ ;
where
P w′ =
Pw
ρw gH
and
0 < z′ < 1
(3.12)
z
H
(3.13)
z′ =
The nonwetting phase pressure distribution is more complex. When we increase
the bottom boundary pressure in steps, the nonwetting phase will infiltrate further into the domain. For each nonwetting phase bottom boundary overpressure
∆P , a particular static equilibrium pressure and saturation distribution can be
derived. A distinct infiltration front is present as long as the nonwetting phase
has not reached the top of the sample. Below the infiltration front, the pressure
distribution is given by:
Pn (z) = ρw gH + Pd + ∆P − ρn gz;
0 ≤ z ≤ zf
(3.14)
3.3. Model description
27
where Pd is the entry pressure of the porous medium, ρn is the nonwetting
phase density, and zf is the height of the infiltration front. The height of the
infiltration front zf is given by:
zf =
∆P
(ρn − ρw )g
(3.15)
Making Equation 3.14 dimensionless, we obtain:
Pn′ (z ′ ) = 1 + Pd′ + ∆P ′ − ρ′ z ′ ;
0 ≤ z ′ ≤ zf′
(3.16)
where
Pn′ =
Pn
;
ρw gH
Pd′ =
Pd
;
ρw gH
∆P ′ =
∆P
;
ρw gH
ρ′ =
ρn
;
ρw
zf′ =
zf
∆P ′
= ′
H
ρ −1
(3.17)
In the remainder of this paper, the prime sign will be omitted, but it must be
understood that all variables are dimensionless unless stated otherwise. Above
the infiltration front, in case of primary drainage, the nonwetting phase is not
present. However, the simple average pressure averages over the whole domain
and therefore a pressure distribution needs to be assumed. Here we assume that
when a phase is not present, its pressure still is defined based on the BrooksCorey relationship. According to the Brooks-Corey relationship, at a wetting
saturation of one, the difference between the nonwetting and wetting phases
(i.e. capillary pressure) must be equal to the local entry pressure. This is a
common assumption in most numerical models (e.g. White and Oostrom, 1997;
Helmig, 1997). The dimensionless pressure distribution for the simple average
above the infiltration front thus is given by:
Pn (z) = 1 − z + Pd ;
zf < z ≤ 1
(3.18)
The saturation distribution can be derived from the nonwetting and wetting
phase pressure distribution through the Brooks-Corey relationship:
Sw (z) = (1 − Swr − Snr )
Pn (z) − Pw (z)
Pd
−λ
+ Swr ;
zf ≤ z ≤ 1
(3.19)
28
Chapter 3. What is the correct definition of average pressure?
Table 3.1: Brooks-Corey parameters and density ratio.
Brooks-Corey parameters
Dimensionless entry pressure, Pd′
Pore size distribution index, λ
Residual wetting phase saturation, Swr
Residual nonwetting phase saturation, Snr , primary drainage
Residual nonwetting phase saturation, Snr , main drainage
Phase density
Density ratio, ρ′
Value
1.53
6.11
0.10
0
0.2
Value
1.6
in the region above the front the wetting saturation is defined as:
Sw (z) = 1;
zf ≤ z ≤ 1
(3.20)
where λ is the pore size distribution coefficient, Swr is the irreducible wetting
phase saturation, and Snr is the residual nonwetting phase-saturation. Table
3.1 contains the values for the different parameters used in this paper. From the
static equilibrium pressure and saturation distributions, average values can be
determined using the averaging operators introduced in Section 3.2. This can
be done for a range of values of ∆P . Note that for the one-dimensional dimensionless domain, integration is not performed over volume V but from zero to
one.
3.4
3.4.1
Results and discussion
The correct average pressure
Before comparing the different averaging operators, we need to determine what
the correct average pressure should be. This can be done by considering the
phase potentials:
Φn (z) = Pn + ρz
(3.21)
Φw (z) = Pw + ρz
(3.22)
Because static equilibrium is assumed, the phase potentials must be constant
throughout the domain, and equal to their boundary values. These constant
3.5. Average pressure during primary drainage
29
potentials at z=0 are found to be equal to:
Φn = 1 + Pd + ∆P
(3.23)
Φw = 1
(3.24)
and
The average of a constant local potential should simply be equal to the local
potential:
< Φn >=< Pn > +ρ < z >= Φn
(3.25)
< Φw >=< Pw > + < z >= Φw
(3.26)
In our dimensionless domain, <z> is equal to 1/2. Based on Equations 3.21,
3.22, 3.24, 3.23, 3.25, 3.26, the correct average pressures are given by:
1
< Pn >=< Φn > −ρ < z >= 1 + P d + ∆P − ρ
2
and
< Pw >=< Φw > − < z >=
1
2
(3.27)
(3.28)
In the remainder of this paper, these average pressures are referred to as the
potential-based average pressures. They represent the correct average pressures
because:
1. they preserve the local constant phase potentials at the upscaled level, and
2. they represent the pressure at the centroid of the domain, <z> .
3.5
Average pressure during primary drainage
In this section, the averaging operators introduced in section 3.2 are compared
to each other and to the potential-based average pressure for primary drainage.
We consider the simple case where throughout the primary drainage process, the
local nonwetting phase pressure is larger than the local wetting phase pressure
when extrapolated in a linear fashion above the infiltration front. The resulting
pressure distribution for one particular situation in the primary drainage process
(∆P = 0.25) and the corresponding average pressures are plotted in Figures 3.3
a, b, c, d, e and Figure 3.4 a.
30
Chapter 3. What is the correct definition of average pressure?
a
b
1
1
Pw
Pn
<Pw>
<Pn>
centroid of domain
extrapolated Pn
0.9
0.7
0.6
0.6
0.5
0.4
0.4
0.3
0.2
0.2
0.1
0.1
3
0
0
Pw
Pn
<Pw>sp
<Pn>sp
centroid of domain
Pw
0.9
Pn
0.8
<Pw>
0.7
0.6
0.6
0.5
0.4
cent of dom
0.4
0.3
0.2
0.2
0.1
0.1
3
<Pn>
0.5
0.3
1
2
Pressure P [−]
s
s
0.7
0
0
3
1
Height z [−]
Height z [−]
0.8
1
2
Pressure P [−]
d
c
1
0.9
centroid of domain
0.5
0.3
1
2
Pressure P [−]
<P >i
n
0.7
0
0
w
0.8
Height z [−]
Height z [−]
0.8
Pw
Pn
i
<P >
0.9
0
0
1
2
Pressure P [−]
3
Figure 3.3: Pressure and saturation distributions used to obtain average pressures for ∆P = 0.25 and ρ = 1.6; a) potential-based average pressure; b) intrinsic phase-volume average pressure; c) simple phase average pressure; d) simple
average pressure.
3.5. Average pressure during primary drainage
e
f
1
0.9
0.8
1
Pw
Pn
[Pw]1
[Pn]1
centroid of domain
extrapolated Pn
0.9
0.8
0.7
0.6
Height z [−]
Height z’ [−]
0.7
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
31
1
2
Pressure P’ [−]
3
0
0.4
0.6
0.8
Water saturation S [−]
1
w
Figure 3.4: Pressure and saturation distributions used to obtain average pressures
for ∆P = 0.25 and ρ = 1.6; a) centroid-corrected phase average pressure; b)
saturation distribution.
In this particular case, the nonwetting phase front has reached approximately
0.4 of the total domain height. As can be seen, the wetting phase averages have
identical values, because the wetting phase is present throughout the domain. An
exception to this is the wetting intrinsic phase-volume average pressure, which
underestimates the average pressure. This is because the local wetting phase
pressures are weighted with the wetting phase saturation, which is lower at lower
elevations where pressure is high. As a result, the lower pressures higher in the
domain are assigned more weight. For a similar reason, the nonwetting intrinsic phase-volume average pressure overestimates the average nonwetting phase
pressure. As can be seen from Figure 3.3b as well, the centroid of the wetting
phase, to which the wetting intrinsic phase-volume average pressure is assigned,
lies above the centroid of the domain. Similarly, the centroid of the nonwetting
phase lies far below the centroid of the domain. All averaging operators, except
the centroid-corrected phase average pressure, give a different nonwetting phase
average pressure than the potential-based average pressure. Figures 3.3 a, b, c,
d, and Figure 3.4 only show the difference between the average pressures for one
particular average saturation (i.e. one moment in the drainage process). To see
how the average pressures evolve through drainage steps, the average pressures
32
Chapter 3. What is the correct definition of average pressure?
4
3.5
3
<Pn>
pressure P[−]
<Pn>i
2.5
<Pn>sp
<Pn>s
2
[Pn]1
<Pw>
<Pw>i
1.5
<Pw>sp
Pw>s
1
[Pw1]
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Average water saturation <Sw> [−]
0.8
0.9
1
Figure 3.5: Average pressures as a function of average saturation for primary
drainage.
are plotted as a function of average saturation in Figure 3.5. The differences
in average pressure for the different averaging operators become smaller as the
drainage process progresses. The largest differences between averaging operators
are found for the nonwetting phase, because is initially not present everywhere
in the domain. For the wetting phase, the average pressures are all equal to
each other and constant throughout the drainage process, except the intrinsic
phase-volume average pressure. Figure 3.6 shows the centroids of the phases and
the centroid of the domain as a function of average water saturation. Clearly,
the centroid of the nonwetting phase moves up as the nonwetting phase front
invades the domain. The centroid of the nonwetting phase does not coincide
with the centroid of the domain even when the nonwetting phase reaches the
outflow of the domain (which occurs at < Sw >= 0.48), because the saturation
continues to be non-linear. Only once the nonwetting phase saturation becomes
uniform, the phase centroid reaches the middle of the domain. The centroid
of the wetting phase coincides initially with the centroid of the domain, as the
domain is completely filled with water. However, during the drainage process,
the wetting phase centroid moves up because the nonwetting phase replaces the
wetting phase from below. At higher elevations, the water saturation is higher
3.5. Average pressure during primary drainage
33
and as a result the wetting phase centroid is situated above the middle of the
domain. The maximum height of the wetting phase centroid corresponds to the
nonwetting phase reaching the top of the domain. After that, the wetting phase
distribution becomes more and more uniform and the wetting phase centroid approaches the centroid of the domain. Now we return to the two requirements for
1
centroid of domain, <z>
nonwetting phase centroid, <zn>i
wetting phase centroid <zw>i
0.9
0.8
Height z’ [−]
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Average water saturation <Sw> [−]
0.8
0.9
1
Figure 3.6: Centroid of the domain and centroids of the phases as a function of
average water saturation < Sw > during primary drainage.
a correct pressure (at static conditions) described in section 3.4.1. The average
potentials are calculated according to:
< Φn >a =< Pn >a +ρ < zn >a
(3.29)
where the superscript a refers to one of the averaging operators. The average coordinate < z >a is calculated in the same manner as the average pressure,
for instance, the intrinsic phase-volume average coordinate is defined as:
Z
1
< zα >i = R
Sα zdV ;
α = w, n
(3.30)
V
V Sα dV
Note that < zα >i can have different values for the wetting and the nonwet-
34
Chapter 3. What is the correct definition of average pressure?
ting phase. Thus, a corrected average operator is the one that, after calculating
< Pn >a and < zn >a and substituting these into Equation 3.30, gives an
average potential that is equal to the local constant potential. In addition ,
< zn >a should be equal to the centroid of the domain, which in this case is
0.5. The potential-based average operator fulfils these requirements, as does the
centroid-corrected phase volume average operator. The other averaging operators however do not meet both requirements. The intrinsic phase-volume average
operator does give an average potential that is equal to the local constant potential, but the average coordinate is not at the middle of the domain. The
same holds for the simple phase average operator. The simple average operator,
although being located at the middle of the domain, does not give the correct
average potential. The intrinsic phase-volume average and simple phase average
can be corrected such that the correct average pressure is obtained. Consider
that for any averaging operator, preserving the local constant potential, the
following holds:
< Φα >a =< Pα >a +ρ < zα >a
(3.31)
Moreover, all the average potentials are equal to the constant local potential
and therefore to each other:
< Φα >i =< Φα >sp =< Φα >= Φa
(3.32)
Using these relations, it is possible to rewrite the average pressures into each
other. For the intrinsic phase-volume average pressure, we obtain:
< Pα >=< Pα >i +ρ < zα >i − < z >
(3.33)
As this equation shows, the correct pressure is obtained by correcting the intrinsic phase-volume average pressure for the difference between the centroid of
the average coordinate < zα >i , and the centroid of the averaging domain <z>.
Actually, this is also exactly what the centroid-corrected phase average does.
Equation 3.34 is a special case of Equation 5.12, where:
∂
∂z
∂
∂z
hPα ii
hzα ii
= −ρ
(3.34)
3.6. Average pressure during main drainage
35
A similar expression can be obtained for the simple phase-average pressure. In
fact, the correct pressure can be obtained from any averaging operator preserving the local constant potential using expressions similar to Equation 3.34.
The simple phase average cannot be corrected in this way, because it does not
preserve the local constant potential and therefore requirement 3.33 cannot be
satisfied. The relation between the intrinsic phase-volume average pressure and
the potential-based average pressure pressure given in Equation 3.34 is only valid
for equilibrium conditions. However, the definition given by Nordbotten et al.
(2008), which was called centroid-corrected average pressure in this paper, holds
for non-equilibrium too. Therefore it is the preferred averaging operator to use.
3.6
Average pressure during main drainage
During primary drainage, the nonwetting phase initially is not present everywhere. As a result, the centroids of the phases are different from each other
and from the centroid of the domain. This difference is largest early in the primary drainage process. Considering a main drainage process on the other hand,
where the nonwetting phase is already present throughout the domain (though
at residual saturation), the difference between the centroids of the phases and
the centroid of the domain are expected to be less. The pressure distributions in
this case are equal to the pressure distributions introduced in Equation 3.12 and
Equation 3.16 for primary drainage. However, now above the infiltration front
the nonwetting phase is already present at residual saturation. Therefore the
choice of the nonwetting phase pressure distribution above the front is important for all averaging operators. We have two choices for the nonwetting phase
pressure distribution above the infiltration front during main drainage. The first
choice is to assume that the nonwetting phase above the infiltration front is
residual but continuous, such that the pressure distribution can be linearly ex2
trapolated from the pressure below the front (see Figure 3.7, Pn0
). Alternatively,
if we assume that the nonwetting phase is discontinuous above the infiltration
front, we may choose anything for the nonwetting phase pressure. For instance,
we may choose that the nonwetting phase pressure above the infiltration front
can be anything in between Pw and Pw + Pd . Here we assume the nonwetting
1
phase pressure above the front is equal to Pw + Pd . (see Figure 3.7, Pn0
).
Figure 3.8 shows the obtained average pressures as a function of average
1
saturation, assuming the nonwetting phase pressure distribution Pn0
. The difference between the different average pressures is now significantly less than that
it was for primary drainage (compare with Figure 3.5). In fact, the nonwetting
simple phase-average pressure and the nonwetting centroid-corrected phase av-
36
Chapter 3. What is the correct definition of average pressure?
1
0.9
0.8
0.7
Height z [−]
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
Water saturation Sw [−]
0.8
1
Figure 3.7: a) Choice of pressure distribution above infiltration front. The shaded
1
area corresponds to Pn0
, which can be anything between Pw and Pw + Pd . How1
ever, here we assume Pn0
= Pw +Pd above the infiltration front; b) Corresponding
saturation distribution.
erage pressure now are equal to the nonwetting simple average pressure. This
is of course because now the same pressure distribution and the same averaging
domain are used in these averaging operators. The intrinsic phase-volume average pressure still is different from the other averages because it weights pressure
1
with saturation. Note that when pressure distribution Pn0
is used, the local
potential is not constant anymore, because the pressure distribution has become
non-linear (there is a kink at the infiltration front). This is further discussed
in the next section. For the wetting phase average pressure, there is not much
change because the pressure distribution stays the same (i.e. hydrostatic) for
2
both primary and main drainage. Next consider Pn0
, the second nonwetting
phase pressure distribution suggested in Figure 3.7. The resulting average pressures are plotted as a function of average saturation in Figure 3.9. Again, the
wetting phase pressures are similar to the primary drainage case. However,
now the nonwetting centroid-corrected phase average pressure is equal to the
potential-based average pressure, as it is for primary drainage. This is because
the assumed nonwetting phase pressure distribution for main drainage is equal
to the nonwetting phase pressure when it is extrapolated to the centroid of the
3.6. Average pressure during main drainage
37
4
<Pn>
<Pn>i
<Pn>sp
<Pn>s
[Pn]1
<Pw>
<Pw>i
<Pw>sp
Pw>s
[Pw1]
3.5
Pressure P’ [−]
3
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Average water saturation <Sw> [−]
0.8
0.9
1
Figure 3.8: Average pressures as a function of average, for main drainage, as1
suming Pn0
(Pn = Pw + Pd above zf ).
4
<Pn>
<Pn>i
<Pn>sp
<Pn>s
[Pn]1
<Pw>
<Pw>i
<Pw>sp
Pw>s
[Pw1]
3.5
Pressure P [−]
3
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Average water saturation <Sw> [−]
0.8
0.9
1
Figure 3.9: Average pressures as a function of average saturation, for main
2
drainage, assuming Pn0
.
38
Chapter 3. What is the correct definition of average pressure?
domain in the case of primary drainage. Moreover, the simple phase average
and simple average pressures now are also equal to the centroid-corrected phase
and potential-based average pressures. The differences between the centroids of
the phases and the middle of the domain are also smaller in the case of main
drainage, as shown in Figure 3.10 (compare to Figure 3.6).
1
centroid of domain, <z>
nonwetting phase centroid, <zn>i
wetting phase centroid, <zw>i
0.9
0.8
Height z’ [−]
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Average water saturation <Sw> [−]
0.8
0.9
1
Figure 3.10: Centroid of the domain and centroids of the phases as a function
of average water saturation < Sw > during main drainage.
3.7
Pn above the infiltration front for main drainage
In the previous section, we assumed two different nonwetting phase pressure dis1
2
tributions above the infiltration front, Pn0
and Pn0
. They gave different results
for several averaging operators. But what is the consequence of using either of
2
these distributions? First, consider the pressure distribution Pn0
, where the
nonwetting phase pressure is extrapolated in a linear fashion from the nonwetting phase distribution below the front (see Figure 3.7). This assumption for the
pressure distribution above the front conserves the local constant potential at
the average scale, because the pressure distribution is linear throughout. At the
same time, this assumption can cause the nonwetting phase pressure (both local
and average) to become smaller than the wetting phase pressure, which results
3.7. Pn above the infiltration front for main drainage
39
in a negative capillary pressure. In the examples discussed above, this did not
happen because for this combination of domain length and boundary pressures,
the (extrapolated) nonwetting phase pressure was larger than the wetting phase
pressure at all times. However, there are situations where the centroid-corrected
capillary pressure (and the potential-based average pressure as well) will become
negative. This occurs for example when the domain is very long, the entry pressure is small, or the density ratio ρn /ρw is large. In other words, this occurs (at
least for some stage during the drainage process) when:
1
∆P + Pd + (1 − ρ) < 0
2
3
2.8
1.5
<Pn>−<Pw>
<Pn>i−<Pw>i
sp
sp
<Pn> −<Pw>
<Pn>s−<Pw>s
[Pn]1−[Pw]1
1.3
1.1
2.4
0.9
2.2
0.7
Pressure P [−]
Pressure P [−]
2.6
2
1.8
0.3
0.1
1.4
−0.1
1.2
−0.3
0.2
0.4
0.6
0.8
Average water saturation <Sw> [−]
1
<Pn>−<Pw>
<Pn>i−<Pw>i
sp
sp
<Pn> −<Pw>
<Pn>s−<Pw>s
[Pn]1−[Pw]1
0.5
1.6
1
0
(3.35)
−0.5
0
0.2
0.4
0.6
0.8
Average water saturation <Sw> [−]
1
Figure 3.11: a) < Pn > − < Pw > vs. < Sw > for the main drainage process,
using nonwetting phase pressure distribution , and a dimensionless entry pressure
of 1.53; b) the same, but with a dimensionless entry pressure of 0.06.
The domain height is implicitly included in the dimensionless variables. Note
that this would occur only when ρ > 1. When ρ < 1, the nonwetting phase immediately occupies the domain when injected from below (as it is lighter than
water). Therefore, there is no infiltration front and little or no differences between the average pressures. However, when infiltration takes place from above,
a front occurs when ρ < 1, and a situation similar to above will occur. Whenever inequality in Equation 3.35 is met, the centroid-corrected phase average
40
Chapter 3. What is the correct definition of average pressure?
and potential-based phase average pressure are smaller than the wetting phase
average pressure:
< Pc >=< Pn > − < Pw >< 0
(3.36)
This occurs for small values of ∆P (how small depends on the values for Pd
and ρ). At some point in the drainage process, ∆P becomes large enough such
that the left hand side of Equation 3.35 becomes larger than zero. Thus, the average capillary pressure would be negative early in the drainage process. This is
illustrated in Figure 3.11, for a situation where the dimensionless entry pressure
Pd was reduced from 1.53 (used in the examples up till now) to 0.06. Reducing
this parameter either means that the entry pressure is decreased, or the domain
height is increased. By increasing the density ratio ρn /ρw , similar results are
obtained. It is evident that for Pd = 0.06, the average capillary pressures are all
negative for large values of < Sw >. Once the infiltration front has reached the
centroid of the domain (at approximately < Sw >= 0.57), the potential-based
and centroid-corrected average capillary pressure become positive again. Al2
though using pressure distribution Pn0
above the infiltration front does conserve
local constant potential, the occurrence of negative capillary pressures when
residual nonwetting phase is present is not physically correct. Next, consider
1
the pressure distribution Pn0
. In this case, the nonwetting phase pressure remains larger than the wetting phase pressure throughout the domain and at all
stages. As a result, capillary pressure would also remain positive. But, now the
local nonwetting phase potential is no longer constant, because the nonwetting
phase pressure distribution is non-linear (as long as the front has not reached
1
the upper boundary). The advantage of using Pn0
is that it gives a physically
acceptable nonwetting phase pressure distribution when that phase is present at
residual saturation. A disadvantage of this assumption arises when two different
(averaging) domains on top of each other are considered (similar to two adjacent
gridblocks in a numerical simulator). Figure 3.12 illustrates such a situation,
where, in a main drainage situation, the infiltration front is beyond the lower
domain, but still somewhere in the upper domain. Thus, the upper part of the
upper block is still at residual nonwetting phase saturation. Table 3.2 gives the
averages calculated for the particular situation of Figure 3.12. When the average
potential for each block is calculated, we find:
< Φn >upper >< Φn >lower
(3.37)
This means that there is a difference in the potential between the two domains
3.7. Pn above the infiltration front for main drainage
41
Table 3.2: Average values for several variables in the two grid blocks shown in
Figure 3.12.
Average values
< Φn >
< Φw >
< Sw >
< Sn >
Pressure distribution
[P n] − [P w]
Lower domain
1.40
1
0.11
0.89
1
2
Pn0
Pn0
0.25
0.25
Upper domain
1.53
1
0.75
0.25
1
2
Pn0
Pn0
0.06
-0.06
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
Height z [−]
Height z [−]
under static conditions, which is a nonphysical result. Concluding this sec-
[Pn] P2n0 upper block
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.5
1
Water saturation Sw [−]
Pw
Pn1
Pn2
[Pw] lower block
[Pn] lower block
[Pw] upper block
[Pn] P1n0 upper block
0
0
0.5
Pressure P [−]
1
1.5
Figure 3.12: a) Pressure distribution and average pressures for two identical
domains placed on top of each other; b) corresponding saturation distribution.
tion, for main drainage, we have two choices for the nonwetting phase pressure
distribution above the infiltration front. Assuming the pressure distribution is
a linear extrapolation of the nonwetting phase pressure below the front, the
constant-potential criterion is satisfied, but under certain conditions (small entry pressure, large domain length, large density ratio) the (extrapolated) local
and average capillary pressures can become negative. When it is assumed that
the nonwetting phase pressure distribution above the front is equal to Pw + Pd ,
42
Chapter 3. What is the correct definition of average pressure?
we obtain physically acceptable local and average capillary pressures, but then
the constant-potential criterion is violated.
3.8
When is the use of a correct operator important?
One could ask the question when the difference between various averaging operators is significant. For instance, when the averaging domain is sufficiently small,
a front will only be present in the very early stages of the drainage process,
and saturation will be more or less constant throughout the domain. In such a
case, all averaging operators will give the same average pressure within a certain range of error. Also, as shown here, when both phases are initially present
(as in main drainage or main imbibition situations), differences among various
averaging operators are negligible. Now, even for the case of primary drainage,
when the difference between the centroids of the phase and the centroid of the
averaging domain becomes sufficiently small, the averaging operators all give the
same result. Thus, we need to develop a criterion in this regard. One possible
criterion is the dimensionless number, Nd :
Nd = ρ(< zn >a − < z >) + (< zw >a − < z >)
(3.38)
When this number is sufficiently small, effects due to choice of the averaging
operator may be neglected. Alternatively, another dimensionless number can be
introduced:
(1 − ρ)êf · êg
Ng =
(3.39)
Pd
where êf is the unit vector in the direction of flow relative to the gravitational
constant, and êg is the gravitational unit vector. When Ng approaches zero
or becomes negative, the difference between various averaging operators can be
neglected, because front formation will not readily occur. But when it becomes
much larger than zero, care should be taken to use the correct averaging operator. Figure 3.13 shows the values of different average pressures as a function
of Ng . Clearly, differences are largest for the nonwetting phase pressure when
Ng > 0. Finally, note that the case discussed in this thesis is a specific case
where a dense nonwetting phase fluid replaces a lighter wetting phase fluid from
below. In this case, a front will exist, and Ng will be positive. The Equations
(such as Equation 3.36) introduced in here only apply to this situation. However,
3.9. Summary and conclusions
43
a similar situation arises when a nonwetting phase fluid, which is lighter than
the wetting phase fluid (i.e. ρ < 1), is forced to replace the wetting phase fluid
from above. In that case, there will be a front as well and Ng will be positive.
Alternatively, consider the cases where a lighter nonwetting phase fluid is forced
into the domain from below, or a denser nonwetting phase fluid is forced into the
domain from above. In these cases, there will be no clear infiltration front, and
Ng is negative, resulting in negligible differences between the average pressures.
4
<Pn>
<Pn>i
<Pn>sp
<Pn>s
[Pn]1
3.5
Pressure P [−]
3
2.5
2
1.5
1
0.5
0
−10
−5
0
5
Ng [−]
10
15
20
Figure 3.13: a) values for the different nonwetting phase average pressures as a
function of Ng.
3.9
Summary and conclusions
In this chapter, several averaging operators are introduced and used to obtain
average or upscaled fluid pressures. The most commonly used averaging operator is the intrinsic phase-volume average, which weights point pressure values
with point saturation values. An alternative is the simple phase average, which
does not weight pressure with saturation but applies an indicator function that
ensures that averaging is performed only over the regions where the phase under
consideration is actually present. Yet another alternative is the simple average, assuming that the pressure of phase α is defined everywhere in the domain
44
Chapter 3. What is the correct definition of average pressure?
of interest (even where it does not exist) and performs the averaging over the
whole domain. The centroid-corrected phase average pressure, or in short the
centroid-corrected pressure was introduced by Nordbotten et al.(2008) as a part
of a family of macroscale pressures. This averaging operator corrects the intrinsic phase average pressure for the distance between the centroid of the averaging
volume and the centroid of the phase. The differences among these averaging
operators by applying them to static equilibrium situations in a vertical homogeneous domain is studied. Both primary and main drainage cases are considered,
where the nonwetting phase was injected through the bottom boundary. At
static equilibrium, pressure and saturation distributions were derived for a given
nonwetting phase bottom boundary overpressure ∆P . These distributions were
used in the calculation of average pressure and saturation. An important feature of static equilibrium is that the total phase potential (i.e. pressure plus
gravity potential) is constant over the whole domain. Therefore, its average will
be equal to the same constant. Thus, it was imposed that the average phase
pressure (which is assigned to the centroid of the averaging volume), plus the
gravity potential at the domain centroid must be equal to the constant phase
potential for any given condition. It is found that the intrinsic phase-volume
average pressure, which is commonly employed in averaging studies, results in a
gradient in the total phase potential, i.e. the above criterion is violated. In fact,
only the centroid-corrected operator satisfies this criterion. However, at high saturations, use of the centroid-corrected average can give rise to negative values of
the difference between the average nonwetting and wetting phase pressures. For
main drainage, differences among various averaging operators are significantly
less because both phases are present initially, such that the difference between
the centroids of phases and the middle of the domain are relatively small. Because the nonwetting phase above the infiltration front is present at residual
saturation, it is important to make a choice for the nonwetting phase pressure
distribution above the front. We may either assume that the nonwetting phase
pressure can be extrapolated linearly from the pressure below the front (i.e. continuous phase), or assume that the nonwetting phase pressure above the front is
equal to the wetting phase pressure plus entry pressure (discontinuous phase).
In the first case, the constant-potential criterion is conserved. However, local
and average capillary pressures can become negative and this is a nonphysical
result. In the second case, the constant-potential criterion is violated, but now
local and average capillary pressures are physically acceptable as they do not
become negative.
BIBLIOGRAPHY
45
Bibliography
Ataie-Ashtiani, B., S. Hassanizadeh, M. Oostrom, M. Celia, and M. White
(2001), Effective parameters for two-phase flow in a porous medium with periodic heterogeneities., Journal of Contaminant Hydrology, 49.
Ataie-Ashtiani, B., S. Hassanizadeh, and M. Celia (2002), Effects of heterogeneities on capillary pressure-saturation-relative permeability relationships,
Journal of Contaminant Hydrology, 56.
Bachmat, Y., and J. Bear (1986), Macroscopic modelling of transport in porous
media, pt. 1: The continuum approach, Transport in Porous Media, 1 (3),
213–240.
Bear, J. (1972), Dynamics of Fluids in Porous Media, Dover Publ.
Bear, J., and Y. Bachmat (1986), Macroscopic modelling of transport in porous
media, pt. 2: Applications to mass, momentum and energy transport, Transport in Porous Media, 1 (3), 241–270.
Dahle, H., and M. Celia (1999), A dynamic network model for two-phase immiscible flow, Computational Geosciences, 3 (1), 1–22.
Das, D., S. Hassanizadeh, B. Rotter, and B. Ataie-Ashtiani (2004), A numerical
study of micro-heterogeneity effects on upscaled properties of two-phase flow
in porous media, Transport in Porous Media, 56.
Gielen, T. (2007), Dynamic effect in two-phase flow in porous media: a pore-scale
network approach, Ph.D. thesis, Utrecht University.
Gielen, T., S. Hassanizadeh, M. Celia, H. Dahle, and A. Leijnse (2004), A porescale network approach to investigate dynamic effects in multiphase flow, Proc.
CMWRXV Conference, University of Noth Carolina.
Gielen, T., S. Hassanizadeh, H. Nordhaug, and A. Leijnse (2005), Upscaling
multiphase flow on porous media: from pore to core and beyond, Dynamic
effects in multiphase flow: a pore-scale network approach. Editors: Das, D.B.
and Hassanizadeh, S.M. Springer Verlag.
Gray, W. (1975), A derivation of the equations for multi-phase transport., Chemical Engineering Science, 30.
Gray, W., and C. Miller (2004), Examination of darcy’s law for flow in porous
media with variable porosity, Environ. Sci. Technol, 38.
46
Chapter 3. What is the correct definition of average pressure?
Gray, W., and C. Miller (2007), Consistent thermodynamic formulations for
multiscale hydrologic systems.fluid pressures., Water Resources Research,
43 (W09408, doi: 10.1029/2006WR005811.).
Gray, W., and K. O’Neill (1976), On the general equations for flow in porous
media and their reduction to darcy’s law, Water Resources Research, 12 (2),
148–154.
Hassanizadeh, S., and W. Gray (1979), General conservation equations for multiphase systems: 1. averaging procedure., Advances in Water Resources, 2 (131144).
Helmig, R. (1997), Multiphase flow and transport processes in the subsurface: a
contribution to the modeling of hydrosystems., Springer.
Manthey, S., S. Hassanizadeh, and R. Helmig (2005), Macro-scale dynamic effects in homogeneous and heterogeneous porous media, Transport in Porous
Media, 58.
Neumann, S. (1977), Theoretical derivation of darcy’s law, Acta Mechanica, 25.
Nordbotten, J., M. Celia, H. Dahle, and S. Hassanizadeh (2007), Interpretation of macroscale variables in darcy’s law., Water Resources Research,
43 (W08430, doi:10.1029/2006WR005018).
Nordbotten, J., M. Celia, H. Dahle, and S. Hassanizadeh (2008), On the definition of macro-scale pressure for multi-phase flow in porous media., Water
Resources Research, 44 (W06502, doi:10.1029/2006WR005715).
Quintard, M., and S. Whitaker (1994), Transport in ordered and disordered
porous media i: The cellular average and the use of weighting functions, Transport in Porous Media, 14.
Quintard, M., and S. Whitaker (1994b), Transport in ordered and disordered
porous media ii: generalized volume averaging., Transport in porous media,
14.
Whitaker, S. (1977), Simultaneous heat, mass, and momentum transfer in porous
media: a theory of drying., Adv. Heat Transfer, 13.
White, M., and M. Oostrom (1997), Stomp. subsurface transport over multiple
phases. user’s guide, Tech. rep., Pacific Northwest National Laboratory.
Chapter
4
Materials and Methods
4.1
Introduction
Commonly, in experiments on saturated or unsaturated flow in porous media,
the capillary pressure-saturation curve is determined in columns in which the soil
sample is placed between hydrophobic and hydrophilic membranes. These membranes are used to obtain relatively rapid capillary pressure saturation curves
(or suction-water content curve in the case of unsaturated flow). For example,
in a drainage experiment, involving two immiscible phases like DNAPL (dense
non-aqueous phase liquid) and water, the hydrophilic membrane does not allow
the nonwetting phase to leave the sample. In drainage experiments, after the
nonwetting phase reaches the hydrophilic membranes, it will start to accumulate
on it, inside the soil. Thus, the pressure in the sample will increase, and the saturation of the wetting phase will approach the irreducible saturation. At that
stage, the nonwetting phase disrupts the hydraulic connection between membranes and the wetting phase inside the column and the outside reservoir. This
may significantly affect the distribution of the fluids inside the soil sample. In this
study, in order to investigate the behavior of the capillary pressure-saturation
relationship under equilibrium condition, a series of drainage and imbibition experiments were carried out. The set-up was designed to measure pressures and
saturations of the two fluids inside the sand as well as external pressure and
average saturation. Moreover, no hydrophilic and hydrophobic membranes were
used, instead a recirculation system was designed. In this chapter, the experimental set-up including devices for measuring local pressures and saturations
as well as average saturation over the whole sample, calibration curves, sample
47
48
Chapter 4. Materials and Methods
preparation, and the experimental procedure are presented.
4.2
Experimental set-up
Tetrachloroethylene (PCE) and de-mineralized, de-aired water were used as the
nonwetting and wetting phases respectively. The physical properties of these
two fluids are given in Table 4.1. In order to visualize the displacement of one
phase by the other, PCE was colored with Sudan-Red dye (1,3 mg/l).
Table 4.1: Fluids properties at 20◦ C.
Properties
Density
Viscosity
Water
1000
1 · 10−3
PCE
1623
0.9 · 10−3
Unit
[kg/m3 ]
[Pa s]
Figure 4.1: Experimental setup (not to scale).
4.2. Experimental set-up
49
Figure 4.2: Sand column set-up.
Figure 4.3: On the left: fluid displacement throughout drainage experiment; on
the right: imbibition experiment.
50
Chapter 4. Materials and Methods
A schematic representation of the experimental set-up is shown in Figure
4.1. It consists of a column 21 cm high with an inner diameter of 9.83 cm (see
Figure 4.2), connected to inflow and outflow burettes 120 cm high and 4.95 cm
in inner diameter. Due to chemical nature of the PCE a Polymethylmetacrilate
(PMMA) was chosen as suitable material for the sand column and burettes, and
viton o-rings were used to avoid any leakage. Although these materials have
high resistance to PCE, the Polymethylmetacrilate if continuously in contact
with such a fluid can change its physical properties and be damaged. For this
reason, various part of the set-up were substituted with new ones as soon as
any visible change on the surface of the plexiglass was observed (e.g. inflow and
outflow burettes were changed every four months). Two stainless-steel porous
plates (pore size +/− 40 µm and 3 mm thickness), one at the base and one
at the top of the column, kept the sand in place. A plate of polyoxymethylene
(PMO) with circular and radial channels (0.5−1 cm wide and 0.3−0.6 cm depth)
underneath the bottom porous plate enabled uniform inflow of the invading fluid
(see Figure 4.4).
Figure 4.4: On top: holder with circular and radial channels; Below: stainlesssteel porous plate.
To keep a desired constant inflow pressure at the bottom of the column, a
pressure transducer was installed in a small reservoir just below the porous plate.
This was connected to a pressure regulator placed at the top of the inflow burette
which regulated the pressure of the gas phase inside the inflow burette. In this
way, any pressure change at the bottom of the column was rapidly and automat-
4.2. Experimental set-up
51
ically compensated by a pressure change in the gas phase, restoring the bottom
pressure to a preset value (see Figure 4.1). The outflow reservoir at the top of the
column was kept at constant atmospheric pressure. This was achieved through
a small opening (0.5 cm diameter) that was connected to a balloon filled with
Argon gas. This prevented any evaporation of water or volatilization of PCE
and dissolution of oxygen in the water phase. The water and PCE pressures at
various location in the column were measured by using selective pore pressure
transducers (Kulite XTM190). The transducers were modified such that they
were either hydrophobic or hydrophilic. Three pairs of pore pressure transducers
were inserted diametrically opposed at three different elevations z1 , z2 , z3 along
the column. Each pair consisted of a wetting phase transducer and a nonwetting phase. A description of the transducer, calibration of the pressure devices
and the preparation procedure for reading of each phase pressure are described
later in this chapter (see section 4.3). The local water saturation was measured
inside the sand column (at the same three elevations as the pore pressure transducers) by using a time domain reflectometry system. The system consists of
TDR probes (see Figure 4.8) connected by 1 m long coaxial cable 50 Ω to a time
domain reflectometry measuring device, TDR100 (Campbell Scientific Inc.), via
coaxial multiplexer units SDMX50 (Campbell Scientific Inc). The TDR100 was
connected to a PC via RS232 port. An algorithm was written in MatLab to
control the multiplexer and to acquire the volumetric water content as a function of time. The design of the TDR probe and calibration are described later
in this chapter (see section 4.4.1). The average fluid saturations over the whole
sand sample was determined from the change in volume of the fluids in inflow
and outflow burettes. This change in volume was quantified by measuring the
change in fluid pressure in the burettes. For this purposes, four differential pressure transducers (PDCR4160) were used, two placed in the inflow burette (dP1
and dP2 ) and two in the outflow burette (dP3 and dP4 ) (see Figure 4.1). The
differential pressures measured the change of pressure head of the two immiscible fluids. As there was no evaporation of water and/or volatilization of PCE,
the volume loss of either of the two phases in the two burettes was assumed to
be inside the sand column. Additional information on the calculation used to
determine the average fluid saturation is presented in section 4.6. The inflow
and outflow burettes, and the water and PCE pumps played an important role
in the design of the experiment, as they are part of a recirculation system of
the nonwetting and wetting phase fluids. Preliminary simulation of drainage experiments showed that when the PCE breakthrough occurred, a large volume of
wetting phase was still present in the column. Thus, in order to reduce the water
saturation, a recirculation system was developed which guaranteed a continuous
52
Chapter 4. Materials and Methods
flow of the nonwetting phase through the sand column, displacing the water until
steady state could be reached. At steady state, there was no flow of water but
there was a steady flow of PCE. A similar recirculation was applied during imbibition. In the traditional drainage experimental set-up, the irreducible water
saturation was achieved by the presence of hydrophobic membrane, that kept
the nonwetting phase in the sample but allowed the wetting phase to flow out of
the sample. During drainage processes, the nonwetting fluid was injected from
the lower side of the column to avoid any instability due to gravity. PCE, after exceeding the oil-entry pressure flowed upward in the sand column displacing
water. The initial displaced water first and PCE (after breakthrough), were both
collected in the outflow burette. Due to higher density the PCE sank to the bottom of the burette and was pumped back to the inflowing burette for continuous
recirculation. In imbibition processes water displaced PCE. After breakthrough
the water was collected in the outflow burette and pumped back to the inflowing
burette. Both water and PCE pumps were controlled by a switch on/off system
which consisted of a couple of magnetic sensors activated and deactivated by a
contact with an aluminium floater inside the outflow burette. One floater and a
couple of magnetic sensors were placed on the lower side of the outflow burette
to allow the switching on and off of the PCE pump throughout the drainage
processes. An other floater and couple of magnetic sensors were mounted on the
top side of the burette allowing the switch to turn on and off the water pump
during imbibition processes. In the inflow burette, a floater was placed on the
surface of water to prevent contact with the gas phase and therefore reduce the
possibility of dissolution of air in water. The solubility of air in water increases
with the increases of pressure. This could have caused a problem, especially during imbibition where water was continuously recirculated. The gas dissolved in
water at high pressure could come out of dissolution as a results of a decrease of
pressure (e.g. with water flowing from the bottom to the top of the column) and
the gas could be trapped in the pore voids, causing changes in relative permeabilities. This situation is explained in section 4.7. All sensors were connected
to a panel box of 12 channels (6 pins connection) and to a power supply with a
constant voltage of 10 Volt and connected to a datalogger controlled by a PC.
The experiments were conducted in a temperature controlled room at 20◦ C.
4.3
Pore Pressure transducers
The pressure transducers are differentiate to read water and PCE phases by mean
of selective filters and membranes. Figure 4.5 shows a section through a selective
pressure transducer. It consists of: a transducer (A); a transducer case (D) in
4.3. Pore Pressure transducers
53
which the transducer device is mounted; either hydrophobic or hydrophillic filter
125 µm thick and 1 cm diameter (B); hydrophobic or hydrophillic membrane 0.5
cm thick and 1 cm diameter (C). A viton o-ring was placed in the transducer
case in contact with the filter to avoid any preferential flow. The surface of the
selective pressure transducer in contact with the sand inside the column was 0.7
cm in diameter, while the surface of the sensor device had a diameter of 0.3 cm .
Figure 4.6 shows both hydrophillic and hydrophobic membranes and filters after
an experiment.
Figure 4.5: A, Pore pressure transducer; B, hydrophillic or hydrophobic filters;
C, hydrophillic or hydrophobic membranes; D, transducer case.
4.3.1
Procedure for differential phase pressure reading
Here the procedure for preparing selective pressure transducers for water and
PCE reading is described. Each pressure transducer sensor and transducer case
was first cleaned in alcohol for 12 hours. Hydrophobic filters, hydrophobic membranes and sensor cases were then de-aired in silicon oil in a vacuum for 12 hours.
Hydrophillic filters and hydrophillic membranes were de-aired in de-mineralized
water under vacuum for 12 hours. The assembling of various parts of a selective
pressure transducer was done inside the corresponding fluids, i.e. de-air de-
54
Chapter 4. Materials and Methods
Figure 4.6: Hydrophobic and hydrophillic filters and membranes after experiment.
mineralized water or silicon oil, avoiding any contact of filters and membranes
with air. First the thin filter was inserted inside the transducer case then the
thick membrane. The membrane has a rough side and a smooth side. During assembly care was taken to ensure that the filter was in contact with the
smooth side of the membrane otherwise the roughness can cause undesirable
flow between the membrane and the filter.
4.3.2
Calibration and absolute error
All sensors device were first calibrated in water to a maximum water head of
10 meter at 20.7◦ C and atmospheric pressure of 1009.4 mbar before mounting.
Figure 4.7 shows the pressures measured versus output voltage in mV/V. Table
4.2 shows the maximum absolute error of each pressure transducer.
Table 4.2: Absolute maximum pressure error of the pressure transducer in Pa.
Sensor
PN1
PN2
PN3
PW1
PW2
PW3
serie number
S4673
C4658
S2801
X3117
Y 1940
X3118
max absolute error [Pa]
66
65
250
97
60
76
4.4. Time Domain Reflectometry
55
100
90
80
Pressure [kPa]
70
60
50
40
30
PN1 = S4673
PN2 = C4658
PN3 = B2891
PW1=X3117
PW2=Y1940
PW3=X3118
20
10
0
−1
−0.5
0
0.5
1
1.5
2
Voltage [mV/V]
2.5
3
3.5
4
Figure 4.7: Calibration curves of pore pressure transducers.
4.4
Time Domain Reflectometry
TDR probes are commonly used to measure water saturation in situ (Topp G.C.
and Annan, 1980; Topp and Reynolds, 1998; Topp and Ferre, 2002; Robinson
et al., 2003; Heimovaara and Bouten, 1990; Heimovaara, 1993, 1994). Their
impedance is related to the geometrical configuration of the probe (size and
spacing of the rods) and also is inversely related to the dielectric constant of
the surrounding material. Thus a change in volumetric water content of the
medium surrounding the probe causes a change in the dielectric constant. This
is seen as a change in probe impedance which affects the shape of the reflected
wave. The shape of the reflection contains information used to determine the
water content. The TDR probe in this setup consisted of two parallel stainless
steel rods 8 cm long and 2 mm diameter, spaced 100 mm center to center and
contained in a holder made of isolating material (Figure 4.8). The probes were
placed horizontally inside the sand, perpendicular to the vertical flow direction.
The region of influence of a TDR depends on the design of the probes and in
particular on the diameter of the rods and the distance between them. For tworods probe, Ferre et al. (2002) calculated the region of influence in a cross-section
perpendicular to the length of the rods to be half of the distance between the
rods. Note that for this specific experimental design, the size of the region of
56
Chapter 4. Materials and Methods
Figure 4.8: Time Domain Reflectometry probe.
influence of the TDR along the vertical flow direction z, is comparable with the
size of the pressure transducers in contact with the sand, this is important for
the compatibility of pressure and saturation measurements. For TDR with a
larger region of influence in the z direction compared with the pressure device,
when a fluid phase moves upwards in the vertical direction, it reaches first the
region of influence of the TDR, which will show a change of saturation, while no
pressure change will be recorded. This caused a mismatch in the measurements
of saturation and pressure at a given position. The determination of water
saturation using the TDR system is based on the analysis of a pulse reflection
from the TDR probe. The pulse generated by the time domain reflectometry
measuring device, TDR100, and its reflection are subject to distortion during
travel between the TDR100 and the TDR probe. The cable connecting the
probe to the reflectometer (TDR100) has a characteristic impedance resulting
in both resistive and reactive losses. Any distortion of the wave form caused by
cable impedance can introduce error into the water saturation determination.
The cable length increases the rise time and the amplitude of the reflection. In
general, water saturation is overestimated with increasing cable length, therefore
a 1 m coaxial cable length from the probe to the TDR100 was considered in the
design of this measurement system.
4.4. Time Domain Reflectometry
4.4.1
57
Calibration of TDR system
TDR’s were calibrated as follows:
θtot ntot = θw nw + θn nn + θs ns
(4.1)
where θtot ntot is the output of the TDRs, θ is the phase content, n is the squared
root of the dielectric permittivity ǫ and the subscripts w, nw and s indicate water, PCE and soil phase respectively.
n=
√
ǫα ;
α = w, n, s
(4.2)
Dielectric permettivity of water, PCE and soil grains are listed in Table 4.3
The voids are occupied by two phase, the wetting and nonwetting phases:
Table 4.3: TDR’s parameters
ǫw
80.36
ǫn
2.5
ǫs
4
θw + θn = φ
(4.3)
and the porosity φ is defined as:
φ=1−
ρb
= 1 − θs
ρs
(4.4)
Substituting Equation 4.3 and Equation 4.4 in Equation 4.1 we obtain:
θtot ntot = θw nw + (φ − θw )nn + (1 − φ)ns
(4.5)
Therefore the water content is determined as following:
θw =
θtot ntot − φnn − (1 − φ)ns
nw − nn
(4.6)
58
4.5
Chapter 4. Materials and Methods
Sand column preparation
The sand used in all of the experiments was a natural Dutch sand called Zeijnzand. It was first sieved such that the grain size was D15 = 0.06 and D60 = 0.09.
It was then washed to flush away fine clay particles. Finally it was dried in a
oven at 105◦ C for 24 hours. The top and the bottom porous plates (see Fgures
4.1, 4.2) were cleaned in a alcohol bath for 40 minutes to remove any impurity, and then in a ultrasonic bath filled with water for another 40 minutes to
remove fine particles which could clog the pores space. De-mineralized water
was de-aired in a distiller. The bottom porous plate was mounted on the holder
plate together with the column and sealed with viton o-ring to avoiding leak.
(see Figure 4.2). Lower pressure transducer, selective pressure transducers and
TDR’s were then mounted in the column, which was subsequently filled with deair de-mineralized water. During this step, care was taken to avoid the contact
of selective pressure transducers with air. Therefore, the water level inside the
column was gradually increased and only when it was near a specific elevation,
the transduce was mounted. The sample packing inside the column was done
continuously, pouring dry sand in water and tapping the column continuously.
With this method compaction of the sample was improved.
4.6
Obtaining saturation column from differential pressures
In the inflow burette, two differential pressure transducers, dP1 and dP2 , were
placed at the lower and at the higher ends. dP1 measures the differential pressure
due to the change in water head, whereas dP2 measures changes in both water
change and PCE.
dP1 = ρw g(hw − hw0 ) + ρa gha − ρa gha
(4.7)
dP2 = ρw g(hw − hn ) + ρn g(hn − hn0 ) + ρa gha − ρa gha
(4.8)
where hw0 and hn0 are the zero starting levels. Combining Equation 4.7 and 4.8
we obtain:
dP2 − dP1 = ρw g(hw0 − hn ) + ρn g(hn − hn0 )
(4.9)
Therefore, the variation of PCE height in the inflow burette can be written as:
hin
n =
dP2 − dP1
(ρn − ρw )g
(4.10)
4.6. Obtaining saturation column from differential pressures
59
Figure 4.9: Inflowing burette, dP1 and dP2 are differential pressures.
Similarly the variation of PCE in the outflow burette is given by:
hout
n =
dP4 − dP3
(ρn − ρw )g
(4.11)
The volume of PCE inside the column is given by the difference of the volume
which enters the column and the volume which leaves the column:
dP2 − dP1 − dP4 + dP3
Abur
(4.12)
Vcol = Vnin − Vnout =
(ρn − ρw )g
where Abur is cross-sectional area of the burette. The water Saturation is calculated as following:
Vcol
Sw = 1 −
(4.13)
VV oid
The total pore volume is:
VV oid = Acol hcol φ
(4.14)
60
4.7
Chapter 4. Materials and Methods
Experimental issues and remarks
It should be mentioned that the actual design of the set-up was not straightforward. The current design was the result of many changes in the original set-up
after encountering various difficulties and shortcomings. One of the major problems was the entrance of air into the sand column during dynamic experiments.
Figure 4.10 is an earlier design of the experimental set-up. It consisted of a sand
column and one burette instead of two as in the final set-up. The burette in
this design served both as inflow and outflow reservoir for the wetting and nonwetting fluids. An important requirement for the experiment was a full control
of the pressures at the top and the bottom boundaries of the sand sample. At
the bottom, the inflowing fluid was injected with a constant pressure (e.g. 20
kPa above atmospheric pressure in dynamic experiment). A constant pressure
was guaranteed by coupling a pressure transducer mounted at the bottom of
the column with a pressure regulator at the top of the burette. Any change in
the inflowing pressure, that was recorded by the transducer caused the regulator
to instantaneously increase or decrease the pressure of the gas chamber inside
the burette such that a constant bottom pressure was guaranteed. At the top
boundary, a small opening connected to a balloon filled with oxygen guaranteed
constant atmospheric pressure. In the imbibition experiment, the column was
initially filled with PCE and water was injected from below to displace PCE.
The PCE displaced by water was pumped back by a peristaltic pump and due
to its high density sank to the bottom of the burette. The water moved upward
as uniform front and after breakthrough was pumped back to the burette and
recirculated for 12 hours. During imbibition processes, after a few hours of recirculation of the water phase, gas bubbles were visible inside the sand. Moreover
its volume increased with time. It should be remarked that the system was deair during the preparation and initially not air was noticed in the sand column.
After a couple of tests, it was found that the presence of air inside the sand was
caused by the dissolution of air in the burette. The solubility of air in water
increases with the increase of pressure. The water which was injected inside the
sand at 20 kPa contained a large volume of dissolved air. As water entered the
column, its pressure decreased (e.g. 20 kPa at the bottom to zero at the top).
This caused the air to come out of solution and to get trapped in the pore space.
The water then flowed out and was pumped back into the burette causing it to
picked up more dissolved air that again came out of solution in the column. This
processes repeated in cycles justified the increase in gas bubbles inside the sand
pore volume with time.
4.7. Experimental issues and remarks
Figure 4.10: Original experimental setup.
61
62
Chapter 4. Materials and Methods
Bibliography
Ferre, P. A., H. H. Nissen, and J. Simunek (2002), The effect of the spatial sensitivity of tdr on inferring soil hydraulic properties from water content measurements made during the advance of a wetting front, Vadose Zone Journal,
1.
Heimovaara, T. (1993), Design of triple-wire time domain reflectometry probes
in practice and theory, Soil Sci.Soc.Am.J., 57 (1410-1417).
Heimovaara, T. (1994), Frequency domain analysis of time domain reflectometry
waveforms: 1. measurement of the complex dielectric permittivity of soils
”
Water Resources Research, 30.
Heimovaara, T., and W. Bouten (1990), A computer-controlled 36-channel time
domain reflectometry system for monitoring soil water contents, Water Resour.Res., 26.
Robinson, D., S. Jones, J. Wraith, D. Or, and S. Friedman (2003), A review of
advances in dielectric and electrical conductivity measurements in soils using
time domain reflectometry, Vadose Zone Journal, 2.
Topp, G., and T. Ferre (2002), Water content, in Methods of Soil Analysis. Part
4. (Ed. J.H. Dane and G.C. Topp), SSSA Book Series No. 5. Soil Science
Society of America, Madison WI.
Topp, G., and W. Reynolds (1998), Time domain reflectometry: a seminal technique for measuring mass and energy in soil, Soil Tillage Research, 47.
Topp G.C., J., Davis, and A. Annan (1980), Electromagnetic determination of
soil water content: measurements in coaxial transmission lines., Water Resources Research, 16.
Chapter
5
Local and average capillary
pressure-saturation
relationships1
5.1
Introduction
A common procedure used to obtain the capillary pressure-saturation relationship for a porous medium is through measure the pressures of the wetting and
nonwetting phases in the fluid reservoirs connected to the sample. The capillary
pressure is then assumed to be given by the difference in the pressure of the
nonwetting phase reservoir and the pressure of the wetting phase reservoir. The
capillary pressure measured in this way is generally considered as the average
capillary pressure representative of the whole sample. Liu and Dane (1995) have
showed that this method can yield inaccurate results due to gravity effects. The
significance of the effect depends on certain combinations of fluid phases and
porous media, the pressure cell height, the position of the measurement sensors,
and the relative density difference of the fluids. Thus, the outcome of the common procedure is not really a capillary pressure and cannot characterize average
capillary pressure-saturation relationship. In this work it is referred to this as
external pressure difference. Instead, fluids pressures and saturation are measured at a number of positions along the column. Then a rigorous averaging of
1 Prepared for submission to Transport in Porous Media, S.Bottero, S.M.Hassanizadeh, P.J.
Kleingeld
63
64
Chapter 5. Local and average capillary pressure-saturation relationships
the internal pressures is used to determine average capillary pressure.
5.2
Steady-state experiments
A series of experiments were carried out under equilibrium conditions in a
relatively homogeneous sand column. These consisted of a cycle of primary
drainage, main drainage and main imbibition processes. The experimental setup is given in section 4.2. Throughout the drainage experiments (primary and
main drainage), PCE was injected into the sand column from below, by increasing its pressure in small increments (typically 300 Pa). At each pressure step,
the system was allowed to equilibrate over a 24 hours period before increasing
the pressure by the next increment. It must be noted that the time interval
of 24 hours was kept throughout the entire experiment. However, preliminary
simulations showed that such a long period was required only when approaching
residual water saturation. At each equilibrium step, local pressures and saturations of both phases as well as the differential pressures in the inflow and outflow
burette were recorded. A primary drainage experiment started with the column
fully saturated with the wetting phase, i.e. water. Initially, the pressure of the
nonwetting was increased until the entry pressure of the sand was reached. At
that point, the nonwetting phase (PCE) flowed upwards into the sand column as
a front entering only the lower portion of the column. An equal volume of water
exited the column at the top. Flow of both liquids decreased dramatically after a
few hours and 24 hours were waited to ensure equilibrium was established. At the
end of each measurement step, the wetting phase reached hydrostatic condition.
The nonwetting phase also reached static conditions in the stage before breakthrough. After breakthrough of the nonwetting phase equilibrium was reached
when water was stagnant at hydrostatic and PCE continued flowing through the
column at constant flow rate. Water displaced by PCE, and PCE that flowed
out of the column after breakthrough, were collected in the outflow burette and
pumped back into the inflow burette. Usually, a primary drainage experiment
is followed by a main imbibition experiment, where the wetting phase displaces
the non-wetting phase. However, in this specific case, at the end of the primary
drainage, due to a defect in the recirculation system, the sand column, was
flushed at ones by water eventually approaching the residual PCE saturation.
That formed the starting condition for main drainage. Thus, at the beginning of
the main drainage experiment, the sand column was filled mainly with water and
partially by PCE at residual saturation. The procedure for the main drainage
experiment was the same as the primary drainage explained above. A main
imbibition experiment was carried out after the main drainage experiment. At
5.3. Calibration procedure
65
that point, the column was mainly filled with the nonwetting phase and water
was at irreducible saturation. Imbibition was started by decreasing the pressure
of the nonwetting phase at the bottom of the column in small increments and
at the same time providing water continuously from above. The water phase
was pumped with a constant velocity from the outflow burette to the top of the
column. It should be reminded that the top of the column was at atmospheric
pressure and there was an overflow for water. Therefore, the water provided at
the top that did not enter the sand column flowed back to the outflow burette.
Once the PCE pressure in the column became low enough, water entered the
column, displacing PCE. This method prevented any instability caused by the
density difference between the two phases. The PCE that flowed out of the
column was collected in the outflow burette and pumped back into the inflow
burette.
5.3
Calibration procedure
The pressure sensors were calibrated considering linear pressure distribution of
the wetting and nonwetting phases at static condition (i.e. when there was no
flow). Under this condition the potential of the two-phases (water and PCE)
are expected to be constant along the column. Therefore, the static pressure
distribution of the wetting phase can be calculated as follows:
Φw = ρw gH = Pw (z) + ρw gz = constant
(5.1)
Pw (z) = ρw g(H − z)
(5.2)
thus,
where ρw is the density of the wetting phase, g is the gravitational acceleration , H is the length of the column, and z is the vertical position. For the
non-wetting phase, the potential at the bottom of the column is given by the
external pressure at the bottom of the column as follows:
Φn = P nbot = Pn (z) + ρn gz = constant
(5.3)
Therefore the nonwetting phase pressure at any z position is given by the eternal
pressure applied at the bottom of the column minus the oilstatic pressure to z
66
Chapter 5. Local and average capillary pressure-saturation relationships
position
Pn (z) = Φn − ρn gz = P nbot − ρn gz
5.4
5.4.1
(5.4)
Experimental Results
Results of primary drainage experiments
Figure 5.1 shows the external pressure applied at the inflow, the wetting and
nonwetting phase pressures in the sand column measured at three elevations z1 ,
z2 and z3 versus equilibrium steps, for a primary drainage experiment. The external pressure was increased incrementally, in steps of 400 Pa, until a maximum
pressure of 12800 Pa. When PCE moved as a front but before equilibrium was
reached, water was also pushed out, and therefore its local pressure increased
during flow too. However, once equilibrium was reached the water pressure
decreased again to the hydrostatic value, that is why water pressures given in
Figure 5.1 are almost constat. The difference between Pw1 , Pw2 , and Pw3 is
due to the hydrostatic pressure distribution. During flow, the pressure of the
nonwetting phase at a given sensor increased dramatically as soon as the phase
reached that sensor. Then the internal nonwetting phase pressures increased incrementally following the external pressure. All sensors had the same slope until
step 17, after which the pressure increase ceased. This occurred at the pressure
step where breakthrough of the nonwetting phase occurred. At that point, there
was a switch from a static equilibrium condition to a steady-state equilibrium
condition, with uniform flow of the nonwetting phase. Figure 5.2 shows the local water saturation at equilibrium at three elevations z1 , z2 , and z3 as well as
the average column water saturation calculated from the changes in the inflow
and outflow burettes. Saturations are plotted against equilibrium steps. Water saturation started from 1, because the column was initially fully saturated
with water and decreased to the irreducible water saturations 0.076 at z1 , 0.083
at z2 and to a higher water saturation 0.162 at z3 , at the end of the primary
drainage processes. As expected the greatest water saturation was recorded at
the uppermost sensor (z = z3 ). The equilibrium local capillary pressures at a
given elevation were calculated by subtracting the wetting phase pressure from
nonwetting phase pressure measured at equilibrium. In Figure 5.3, the local capillary pressures at three elevations are plotted versus the local water saturation.
In the same graph, the external pressure versus the average water saturation is
plotted. From the local capillary pressure, the entry pressure Pd is around 4000
Pa, the same entry pressure was found by subtracting the hydrostatic pressure
at the bottom of the column from the nonwetting phase pressure at the time of
5.4. Experimental Results
12000
Pressure [Pa]
10000
8000
67
P external
PN1
PN2
PN3
PW1
PW2
PW3
6000
4000
2000
0
5
10
15
20
Equilibrium Steps [−]
25
30
Figure 5.1: Measured nonwetting and wetting phase pressures versus equilibrium
steps along with the external pressure throughout primary drainage experiment.
1
<Sw>
Sw1
Sw2
Sw3
0.9
Water Saturation, Sw [−]
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
5
10
15
20
Equilibrium Steps, [−]
25
30
Figure 5.2: Measured local water saturations and average water saturation versus
equilibrium steps during primary drainage experiment.
68
Chapter 5. Local and average capillary pressure-saturation relationships
P−Sw external
Pc−Sw at z1
Pc−Sw at z2
Pc−Sw at z3
12000
Capillary Pressure [Pa]
10000
8000
6000
4000
2000
0
0
0.2
0.4
0.6
Water Saturation [−]
0.8
1
Figure 5.3: Local capillary pressure versus local water saturation at different
elevations along with the external pressure versus average saturation throughout
primary drainage.
entry. The entry pressure found in this experiment is in disagreement with that
reported by Hassanizadeh et al. (2005) for the same type of sand. They found an
entry pressure of around 5800 Pa. The capillary pressure-saturation relationship
is assumed to be a property of the medium and fluids. Thus the two experiments
are expected to provide the same parameters. However, it should be reminded
that in their set-up hydrophobic and hydrophillic membranes were placed at
the top and at the bottom of the sample. So the question arises whether these
membranes have an effect on the capillary pressure-saturation relationship. The
locally-measured capillary pressure curves more or less coincide indicating that
the column was reasonably homogeneous. It is clear that the capillary pressure
based on external (top and bottom) reservoirs is significantly larger than the local capillary pressure curves. The difference is mainly due to the effect of gravity
as well as viscous forces when the nonwetting phase flows at steady state.
5.4.2
Results of main drainage and imbibition experiments
Figure 5.4 shows pressures of wetting and nonwetting phases during main drainage
processes. In the same graph, the external pressure of the nonwetting phase at
5.4. Experimental Results
69
16000
P external
Pn1
Pn2
Pn3
Pw1
Pw2
Pw3
14000
12000
Pressure [Pa]
10000
8000
6000
4000
2000
0
−2000
35
40
45
50
Equilibrium Step, [−]
55
60
Figure 5.4: Measured wetting and nonwetting pressures versus equilibrium steps
along with the external pressure throughout main drainage experiment.
1
<Sw>
Sw1
Sw2
Sw3
0.9
Water Saturation, Sw[−]
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
30
35
40
45
50
Equilibrium Steps, [−]
55
60
Figure 5.5: Measured water saturation versus equilibrium steps in main drainage.
70
Chapter 5. Local and average capillary pressure-saturation relationships
P−Sw external
Pc−Sw at z1
Pc−Sw at z2
Pc−Sw at z3
12000
Capillary Pressure [Pa]
10000
8000
6000
4000
2000
0
0
0.2
0.4
0.6
Water Saturation [−]
0.8
1
Figure 5.6: Local capillary pressure versus water saturation along with the external pressure versus average water saturation in main drainage.
the bottom of the column is shown. The pressure was increased incrementally in
steps of 300 Pa until a maximum pressure of 15500 Pa. Similar to the primary
drainage experiment, the local nonwetting phase pressures have more or less the
same trend as the external pressure until step 44 where the system switched
from static equilibrium to steady-state equilibrium condition. One would expect
that the water pressures would remain constant for all equilibrium points. This
was the case for Pw3 sensor but not for sensor Pw1 and Pw2 which decreased
significantly at step 51. This behavior occurred because the water phase became
discontinuous in the lower half of the column as the irreducible water saturation was approached. Because PCE was injected from below, the water phase
in the upper part of the column maintained continuity. This is also evident in
Figure 5.5 where saturation is plotted versus equilibrium steps. Saturation at z3
remained higher than the irreducible water saturation of 0.1.
Figure 5.5 also shows the local saturation at the three elevations z1 ,z2 , and
z3 as well as the average water saturation. Note that the initial water saturation in the main drainage is not equal to unity because initially there is a
residual PCE saturation, Srn . Finally, local capillary pressure-saturation curves
and the external phase pressure difference versus the measured average column
5.4. Experimental Results
71
15000
P external
Pn1
Pn3
Pn3
Pw1
Pw2
Pw3
Pressure, [Pa]
10000
5000
0
−5000
65
70
75
80
Equilibrium step, [−]
85
90
Figure 5.7: Measured local wetting and nonwetting phase pressures along with
the external pressure versus equilibrium steps throughout main imbibition experiment.
1
0.9
Water Saturation, Sw[−]
0.8
<Sw>
Sw1
Sw2
Sw3
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
60
65
70
75
80
Equilibrium Steps, [−]
85
90
Figure 5.8: Measured local water saturation throughout main imbibition.
72
Chapter 5. Local and average capillary pressure-saturation relationships
saturation are plotted in Figure 5.6. Also here as for primary drainage, an entry pressure of around 4000 Pa was found. The local curves coincide quite well
and the external curve shows a much larger capillary pressure values than the
local values. Imbibition experiments were performed at the end of the main
drainage experiment. This was started by decreasing the pressure of the nonwetting phase at the bottom of the column and providing water continuously
at the top. Therefore, the flow was downwards, reversing the drainage process.
Similar to the drainage process, a stable front movement was observed. The
external pressure of the nonwetting phase at the bottom was decreased incrementally from 15000 Pa to 4500 Pa, in small steps of 400 and 200 Pa. Figure
5.7 shows the phase pressures versus equilibrium steps. Wetting phase pressures
measured by sensors showed negative values which was not expected. However,
a hydrostatic distribution was attained when the water phase reached the corresponding wetting phase sensor at step 85, Figure 5.8. The negative pressure was
probably due to discontinuity of the wetting phase along the column. In fact,
as shown in Figure 5.8, the water saturation at the beginning of the imbibition
was equal to irreducible water saturation. Local capillary pressure-saturation
curves and the externally-based capillary pressure curve are plotted in Figure
5.9. Here again, the same trends as for primary and main drainage curves
are observed. Primary drainage, main drainage and main imbibition capillary
pressure-saturation curves, obtained at each measurement point z1 , z2 , andz3 , are
fitted by Van Genuchten model (Van Genuchten, 1980). The capillary pressure
Pc is considered as independent variable and Sw as dependent variable:
Se = [1 + (αPc )n ]m
(5.5)
For primary drainage the effective saturation Se is expressed as following:
Se =
Sw − Srw
;
1 − Srw
Srw ≤ Sw ≤ 1
(5.6)
and for main drainage and main imbibition as:
Se =
Sw − Srw
;
1 − Srw − Srn
Srw ≤ Sw ≤ 1 − Srn
(5.7)
Van Genucthen parameters are optimized by a trust-region algorithm (see Matlab). Fitted parameters α and n as well as the goodness of the fit (R2 ) are
reported in Table 5.1. Figure 5.10 shows the comparison between fitted and
5.5. Average capillary pressure-saturation relationship
73
10000
P external
Pc−Sw at z1
Pc−Sw at z2
Pc−Sw at z3
9000
Capillary Pressure [Pa]
8000
7000
6000
5000
4000
3000
2000
1000
0
0
0.2
0.4
0.6
Water Saturation [−]
0.8
1
Figure 5.9: Local capillary pressure versus water saturation along with the external pressure versus average saturation during main imbibition experiment.
measured Pc − Sw curves. Note that in the parameters optimization the points
in z3 at the lowest saturation in the main imbibition curve are excluded.
Table 5.1: Van Genuchten parameters obtained for primary drainage, main
drainage and main imbibition.
VG parameters
n
alpha
R-square
5.5
Primary Drainage
8.037
0.0002165
0.993
Main Drainage
7.68
0.0002065
0.987
Main Imbibition
5.825
0.0003894
0.994
Average capillary pressure-saturation relationship
In sections 5.4.1 and 5.4.2, local phase pressures and water saturations as well
as capillary pressure-saturation curves inside the domain at different elevations
were presented. The question arises how should average pressures and water
saturations be calculated over the whole domain using local measurements. In
74
Chapter 5. Local and average capillary pressure-saturation relationships
Primary Drainage
Main Drainage
12000
Pc−Sw at z1
Pc−Sw at z2
Pc−Sw at z3
Pc−Sw fitted VG
10000
8000
Capillary pressure [Pa]
Capillary pressure [Pa]
12000
6000
4000
2000
0
0
0.2
0.4
0.6
0.8
Water saturation [−]
8000
6000
4000
2000
0
1
Pc−Sw at z1
Pc−Sw at z2
Pc−Sw at z3
Pc−Sw fitted VG
10000
0
Main Imbibition
1
12000
Pc−Sw at z1
Pc−Sw at z2
Pc−Sw at z3
Pc−Sw fitted VG
10000
8000
Capillary pressure [Pa]
Capillary pressure [Pa]
0.4
0.6
0.8
Water saturation [−]
Fitted Curves VG model
12000
6000
4000
2000
0
0.2
0
0.2
0.4
0.6
0.8
Water saturation [−]
1
Primary Drainage
Main drainage
Main Imbibition
10000
8000
6000
4000
2000
0
0
0.2
0.4
0.6
0.8
Water saturation [−]
1
Figure 5.10: Capillary pressure-saturation curves at elevation z1 , z2 , and z3
along with the Pc − Sw curve fitted by van Genuchten model: a) Primary
Drainage; b) Main Drainage; c) Main Imbibition; d) comparison between Pc −Sw
curves during primary drainage, main drainage and main imbibition fitted by Van
Genuchten model.
5.5. Average capillary pressure-saturation relationship
75
this section, the average pressure and saturation are obtained from the local variables by means of various averaging operators and compared with the measured
average values.
5.5.1
Averaging operators
The average saturation is defined as the total volume of the α phase over the
total pore volume of the column. Assuming constant porosity, it can be written
in a discretized form as follows:
< Sα >=
N
X
Sαj
j=1
N
(5.8)
where N represents the number of observation points. While the average saturation can be simply calculated by this arithmetic average, the definition of
average phase pressure is not straightforward. This issue has been recently discussed by Nordbotten et al. (2007, 2008) and Korteland et al. (2009). Here, four
average pressure operators are used and discussed: simple average, simple phaseaverage, intrinsic phase-average and centroid-corrected average as in Chapter 3.
The simple average pressure is an arithmetic average of the phase pressure. The
discretized form becomes:
PN
j
j=1 Pα
< Pα >s =
;
α = w, n
(5.9)
N
For primary drainage, the nonwetting phase is not always present everywhere
within the sample. Nevertheless, a pressure is assigned everywhere. The pressure of the nonwetting phase, when it is not present, is assumed to be equal
to the wetting phase pressure (i.e. capillary pressure being zero). Usually in
many numerical codes (e.g. STOMP, White and Oostrom (1997); MUFTE-UG,
Helmig (1997)), when the nonwetting phase is not present its pressure is defined
to be equal to the wetting phase pressure plus the entry pressure. However, this
assumption is quite artificial as the capillary is raised to an entry value even in
the saturated domain. One way of avoiding to introduce this assumption is to
average the α-phase pressure only within the region where the phase is actually
76
Chapter 5. Local and average capillary pressure-saturation relationships
present. We refer to this as Simple Phase-Average pressure expressed as follows:
sp
< Pα > =
PN
j=1
Pα j(Sα >0)
N(Sα >0)
;
(5.10)
This means that during primary drainage, whenever there is a front, the average pressures of the two phases as defined in Equations 5.10 are not calculated
over a common domain. When the nonwetting phase front advances along the
domain, the nonwetting phase pressure is averaged only for the region behind
the front which at the early stage of the displacement can be just a small fraction of the whole domain. The water phase, however, is present everywhere, and
its pressure is averaged over the entire domain. Therefore, the centroid of the
nonwetting phase does not coincide with the centroid of the wetting phase or
with the centroid of the column. The traditional way of averaging pressure is the
intrinsic phase average where basically the pressure of the phase is weighted by
the phase saturation. Assuming constant porosity, the average phase pressure
can be written as follows:
PN
j j
j=1 Sα Pα
i
< Pα > = PN
(5.11)
j
j=1 Sα
Here also the two phases are averaged over different domains and, therefore
the centroids of averaging domains are not the same. Nordbotten et al. (2008)
introduced another averaging operator, centroid-corrected phase average which
in one dimensional form may be written as:
[Pα ] =< Pα >i +
∂
1
(z̄− < zα >)
< Pα >i
∇·z
∂z
(5.12)
where z̄ is the point to which [Pα ] is assigned, i.e. the centroid of the column or
the region of interest and it is defined as follows:
z̄ =
1
H
2
(5.13)
where H is the length of the averaging domain, and < zα > is the centroid
of the phase, defined as:
5.5. Average capillary pressure-saturation relationship
PN
j=1
< zα >= PN
Sαj zαj
j=1
Sαj
77
(5.14)
In this averaging operator, the intrinsic-phase average pressure < Pα >i is
corrected for the distance between the centroid of the averaging volume, z̄, and
the centroid of the phase, < zα >. The derivatives of the intrinsic-phase average pressure and position < zα > can be calculated for a vertical column with
upward flow as follows (Korteland et al., 2009):
∂
1
< Pα >i =
− < Pα >i Sαtop − Sαbot + Sαtop · Pαtop − Sαbot · Pαbot
∂z
< Sα > H
(5.15)
∂
1
< zα >i =
− < zα > Sαtop − Sαbot + Sαtop · zαtop − Sαbot · zαbot
∂z
< Sα > H
(5.16)
where the subscripts ’top’ and ’bot’ refer to the values of the variable at the
top and bottom and the averaging domain. In this case, these would be points
z3 and z0 respectively. The average phase pressures and saturation are calculated based on the local measurements at three elevations z1 , z2 , z3 . Moreover
an additional point at z = z0 is considered, which is just above the porous plate
inside the column. The pressure of the nonwetting phase assigned to this virtual
point is obtained from the external nonwetting pressure applied at the bottom
boundary as follows:
Pn0 = Pext − d0 ρn g
(5.17)
where d0 is the thickness of the bottom porous plate. The wetting phase will
have a hydrostatic pressure distribution and, thus, its pressure at z0 is given as
follows:
Pw0 = ρw gH
(5.18)
The water saturation at z = z0 is calculated by using the van Genuchten model
(Van Genuchten, 1980). In Table 5.1 the van Genucthen parameters for primary
as well as main drainage and main imbibition curves are reported. The averaging
78
Chapter 5. Local and average capillary pressure-saturation relationships
domain, therefore, is for z0 to z3 , with a height H = z3 − z0 = 0.187m. The
average capillary pressure is calculated by subtracting the average water phase
pressure from the average nonwetting phase pressure.
5.6
Reference average capillary pressure
In the previous section, various averaging operators were defined. The question
arises which operator provides an average capillary pressure that representative
of the whole domain. An answer to this question can be found for the static stage
prior to breakthrough of the nonwetting phase. If there is no flow, the hydraulic
potential for each phase must be constant along the column. So, the average
hydraulic potential is equal to the potential specify by the local phase potential.
Now, given the fact that an average pressure is assigned to the centroid of the
averaging domain, the average hydraulic potential of the phase is equal to:
1
< Φα >=< Pα > + ρα g(H + d0 )
2
(5.19)
and for the nonwetting phase:
1
< Φn >=< Pn > + ρn gH = Pnbot = Pnext − ρn gd0
2
(5.20)
For the wetting phase the average hydraulic potential is:
1
< Φw >=< Pw > + ρw gH = Pwbot = ρw gH
2
(5.21)
Thus the correct average wetting and nonwetting pressures under static condition can be calculate from Equations 5.20 and 5.21 respectively as:
1
< Pn >= Pnext − ρn gd0 − ρn gH
2
< Pw >=
1
ρw gH
2
(5.22)
(5.23)
These average phase pressures has been presented and discuss in section 3.4
and are referred to as potential-based average pressures. The average capillary
pressure is calculated as follows:
5.6. Reference average capillary pressure
< Pc >=< Pn > − < Pw >
79
(5.24)
Thus we consider the potential-based average pressure as our reference to identify
the correct average operator and the correct average capillary pressure.
5.6.1
Results and discussion
Figure 5.11 shows the average wetting and nonwetting phase pressures versus
equilibrium steps during primary drainage. Significant differences between the
various averaging pressure operators are noted for the nonwetting phase prior to
breakthrough (equilibrium step 17). After the nonwetting phase front reaches the
end of the domain, all average nonwetting phase pressures are in good agreement
with each other. Nonwetting phase pressure based on the simple average pressure
lies below the other curves and below the reference pressure. This is because
by this operator, the nonwetting phase pressure is also averaged over the region
above the front where the nonwetting phase is not present, and its pressure
is set equal to the wetting phase pressure. In the case of simple-phase average
operator, the average nonwetting phase pressure is calculated only in the domain
behind the front. As result, it overestimates the nonwetting phase pressure
reference at the early stage of the front displacement.
For example, when
the front has just entered the column, then the simple-phase averaged pressure
of the nonwetting phase is almost equal to the nonwetting phase pressure at
the bottom of the column (Figure 5.11). Also the nonwetting intrinsic-phase
average overestimates the reference curve because the high nonwetting phase
pressure is weighted by the high nonwetting phase saturation. At the same time
the intrinsic-phase average pressure of the wetting phase is lower than the other
average pressures because higher up in the column, where pressure is low, it is
weighted by the high wetting saturation and at lower elevations, where water
pressure is high, it is weighted by low saturation. The centroid-corrected phase
average pressure as mentioned above, corrects the intrinsic-phase average of
the wetting and nonwetting phases for the distance between the centroid of the
average domain z̄ = H/2 and the centroid of each phase < zα >. This is observed
in Figure 5.12 where the centroids of the wetting and nonwetting phases along
with the centroid of the average domain are plotted versus equilibrium steps. It
is clear that the centroid of the nonwetting phase moves up as the nonwetting
front is at higher position in the column at different equilibrium steps. After
the nonwetting phase breakthroughs and once the nonwetting phase saturation
becomes uniform its centroid reaches the middle of the averaging domain. At the
80
Chapter 5. Local and average capillary pressure-saturation relationships
12000
10000
Pressures [Pa]
8000
n Simple Average
n Intrinsic−Phase Average
n Centroid−Corrected Average
n Based−Potential (reference)
w Simple Average
w Intrinsic−Phase Average
w Centroid−Corrected Average
w Based−Potential (reference)
6000
4000
2000
0
5
10
15
20
25
30
Equilibrium steps
Figure 5.11: Averaged nonwetting and wetting phase pressures versus equilibrium
steps during primary drainage
0.2
centroid of domain
non−wetting phase centroid
wetting phase centroid
0.18
0.16
<z alfa> [m]
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
5
10
15
20
Equilibrium steps
25
30
Figure 5.12: Centroid of the average domain and centroids of the wetting and
nonwetting phases versus equilibrium steps during primary drainage.
5.6. Reference average capillary pressure
81
10000
9000
Capillary Pressure [Pa]
8000
7000
6000
5000
4000
3000
2000
1000
0
0
local fitted by VG model
based Potential
Simple Average
Simple Phase Average
Intrinsic Phase Average
Centroid Corrected Average
0.2
0.4
0.6
Water Saturation [−]
0.8
1
Figure 5.13: Average Pc − Sw curves along with the local one obtained by fitting
Pc − Sw data by Van Genuchten model. All curves refer to primary drainage.
beginning of the primary drainage as the column is filled by water, the centroid
of the water phase is the same as the centroid of the averaging domain. Then,
when the nonwetting phase front moves upwards, the centroid of the wetting
phase moves upwards too, as areas of higher water saturation are found at higher
elevations. At equilibrium step 16 the front reaches the end of the domain and
the centroid of the wetting phase will be at its highest position. Afterwards as
the water saturation also becomes uniform, the water phase centroid moves down
to approach the centroid of the averaging domain. However, because of the high
water saturation present at higher elevations inside the column, the centroid of
the wetting phase differ from the centroid of the averaging domain. In Figure
5.13, the four average capillary pressures are plotted versus the average wetting
saturation < Sw > and compared with the potential-based capillary pressuresaturation curve. The comparison between averaged Pc − Sw curves based on
various averaging operators shows that the centroid-corrected average approaches
better the reference curve or potential-based curve. Note that the potential-based
average capillary pressure is valid only under static condition while the centroid
corrected holds also under flow condition (i.e after breakthrough).
It should be noted that after the front reaches the end of the domain at
82
Chapter 5. Local and average capillary pressure-saturation relationships
12000
10000
n Simple Average
n Intrinsic−Phase Average
n Centroid−Corrected Average
n based Potential
w Simple Average
w Intrinsic−Phase Average
w Centroid−Corrected Average
w based Potential
Pressures [Pa]
8000
6000
4000
2000
0
35
40
45
50
Equilibrium steps [−]
55
60
Figure 5.14: Averaged nonwetting and wetting phase pressures versus equilibrium
steps during main drainage.
0.2
centroid of domain
non−wetting phase centroid
wetting phase centroid
0.18
0.16
<z alfa> [m]
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
35
40
45
50
Equilibrium steps
55
Figure 5.15: Centroid of the average domain and centroids of the wetting and
nonwetting phases versus equilibrium steps throughout main drainage.
5.6. Reference average capillary pressure
83
10000
9000
Capillary Pressure [Pa]
8000
7000
6000
5000
4000
3000
2000
1000
0
0
fit VG model
based Potential
Simple Average
Intrinsic Phase Average
Centroid Corrected Average
0.2
0.4
0.6
Water Saturation [−]
0.8
1
Figure 5.16: Capillary pressure-saturation curves based on different averaging
operators along with the reference curve and the local one. These refer to main
drainage.
Sw = 0.4 water saturation, simple average, simple-phase average and centroidcorrected average lie one above each other. Thus, at low water saturation the
differences between the various averages are not significant. Also in Figure 5.13,
the local capillary pressure-saturation curve is plotted. The comparison between
the capillary pressure-saturation based on the centroid-correct averaging operator with the local one, shows that these two curves are close to each other. Figure
5.14 shows wetting and nonwetting average pressures obtained by employing various averaging operator versus average water saturation during main drainage.
Note that in case of main drainage, both wetting and nonwetting phases are
present in the whole domain, from the start. Therefore, the pressures obtained
by the simple average and by the simple-phase averaging operators are identical.
It is clear that the differences between the average nonwetting pressures are less
pronounced than in the case of primary drainage. This is because both phases
are present from the beginning of the drainage process.
This is also observed in Figure 5.15 where the centroids of both nonwetting
and wetting phases along with the centroid of the averaging domain are plotted
against equilibrium steps. In this case the distances between the centroid of the
average domain and the centroids of the two phases are smaller than in the case
84
Chapter 5. Local and average capillary pressure-saturation relationships
12000
n Simple Average
n Intrinsic−Phase Average
n Centroid−Corrected Average
n based Potential
w Simple Average
w Intrinsic−Phase Average
w Centroid−Corrected Average
w based Potential
10000
Pressure [Pa]
8000
6000
4000
2000
0
−2000
70
75
80
Equilibrium step [−]
85
90
Figure 5.17: Average nonwetting and wetting phase pressures during main imbibition.
0.2
centroid of domain
non−wetting phase centroid
wetting phase centroid
0.18
0.16
<z alfa> [m]
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
70
75
80
Equilibrium steps
85
90
Figure 5.18: Centroid of the average domain along with the centroid of the wetting and nonwetting phase versus equilibrium steps during main imbibition.
5.6. Reference average capillary pressure
85
10000
local fit VG model
based Potential
Simple Average
Intrinsic Phase Average
Centroid Corrected Average
9000
Capillary Pressure [Pa]
8000
7000
6000
5000
4000
3000
2000
1000
0
0
0.2
0.4
0.6
Water Saturation [−]
0.8
1
Figure 5.19: Average capillary pressure-saturation curves obtained by different
averaging operators along with the reference curve and with the local Pc − Sw .
These regard main imbibition process.
of primary drainage, see Figure 5.12. Finally, in Figure 5.16, capillary pressures
based on simple average, intrinsic-phase average and centroid-corrected average are plotted versus average wetting saturation along with the representative
Pc − Sw curve. Also shown in Figure 5.16 is the local main drainage capillary
pressure curve. At Sw < 0.6, the simple average and the centroid-corrected average curves are in good agreement with the reference curve and with the local
curve, while the intrinsic phase average lies just above them. The potentialbased reference curve differs from the other at low average wetting saturation.
However, the comparison between these curves at this low wetting saturation is
not appropriate. In fact the potential based average operator holds only under
no flow condition. In contrast at higher wetting saturation, Sw > 0.6 all average
curves underestimate both the reference curve and the local curve. Figure 5.17
shows average wetting and nonwetting pressures along with the wetting and
nonwetting base-potential pressure. The differences between the average nonwetting pressures obtained by various averaging operators is not pronounced. A
discrepancy is seen between the wetting potential-base average pressure and the
other average wetting pressures. This is due to the fact that at the beginning of
the imbibition process the column is mostly filled by the nonwetting phase with
86
Chapter 5. Local and average capillary pressure-saturation relationships
the wetting phase being at irreducible saturation. Thus, the wetting phase is
discontinuous. The negative value of the average wetting pressures are obviously
due to the negative values observed in local pressures (see Figure 5.7). The applicability of the based-potential averaging operator requires a continuity of the
phase along the domain which is not satisfied for the wetting phase. Thus, in
this case, the comparison with the reference curve is not appropriate to identify
the correct average pressure. Figure 5.18 shows the centroid of the wetting and
nonwetting phases and the centroid of the averaging domain versus equilibrium
steps. The dashed line shows the centroid of the average domain at 0.936 m.
At the time that the wetting front reaches sensor z3 from above, the rest of
the sand column was filled mainly by the nonwetting phase, while the wetting
phase was at irreducible saturation. Therefore, the centroid of the nonwetting
phase is almost the same as the centroid of the averaging domain, while the
centroid of the wetting phase is above because of the large wetting saturation
in z = z3 . As the wetting front moves down into the averaging domain, the
centroid of the wetting phase also moves downward. Once the front reaches the
bottom of the domain the water phase is present everywhere at a high saturation (step 86) and its centroid approaches the centroid of the average domain. In
Figure 5.19, the averaged capillary pressures-saturation curves are plotted along
with the local capillary pressure-saturation curve. For the sake of completeness
also the potential-based average Pc − S are also shown. However, the reader is
reminded that, because of the discontinuity of the water phase along the column, this is not a correct capillary pressure-saturation curve. Thus, it should
not taken in consideration. It will serve to understand what type of error can
eventually occur by adopting this operator under the condition explained above.
At saturations 0.4 ≤ Sw ≤ 0.8, all averaged curves are in good agreement with
each other and with the local curve. Good agreement can be also found at low
water saturation close to irreducible wetting saturation. For wetting saturation
between irreducible and Sw < 0.2 though, all average curves differ from each
other and from the local curve.
5.7
Summary and conclusions
A common procedure to determined capillary pressure is by subtracting the wetting phase pressure from the nonwetting pressure measured in fluid reservoirs external to the sand column. This is then assumed to be the representative average
capillary pressure for the fluids-porous medium system. The capillary pressuresaturation curve derived from external measurements does not correspond to
the capillary pressure-saturation curve representative of the fluid-porous media
5.7. Summary and conclusions
87
system. This method is inaccurate because does not account for the effects of
gravity. Instead, average capillary pressure should be determined from local pressure measurements employing a more rigorous averaging procedure. Drainage
and imbibition experiments were performed under equilibrium condition. Phase
pressures and saturation inside the column as well as external pressure and average saturation were recorded at each equilibrium step. Various averaging operators were considered: simple average , simple-phase average, intrinsic-phase
average and centroid-corrected average. To establish which operator gives the
correct average pressure another operator was introduced named potential-base
average. This operator is based on the hydraulic potential that assumes continuity of a phase along the domain and a linear phase pressure distribution under
no flow conditions. During primary drainage a large difference between nonwetting phase pressures calculated by employing various operators was noticed
whenever a front was present. The simple average operator for definition averaged the nonwetting phase also in the region behind the front where actually its
phase is not present. As consequence, the resulting nonwetting average pressure
was underestimated. The simple-phase operator that averaged the nonwetting
phase only over the region where the phase is present, overestimated the reference pressure in the fist stage of the front development. The intrinsic-phase
average weights the phase pressure by the phase saturation. As a result the wetting and nonwetting phases were averaged over two different averaging domains.
Thus, the centroid of the wetting phase did not coincide with the centroid of the
nonwetting phase and with the centroid of the averaging domain. The centroidcorrected average operator corrects the intrinsic-phase average pressure for the
distance between the centroid of the domain and the centroids of the two phases.
This was found to be in good agreement with the centroid-corrected average. After nonwetting phase breakthrough, both phases were present in the domain and
the differences between pressures obtained by the various average operators were
negligible. In this case the centroid of the each phase was closed to the other and
to the centroid of the averaging domain. The comparison between averaged capillary pressure-saturation curves has shown that the centroid-corrected average
approached the potential-based Pc − S curve. While the potential-based is valid
only under static condition, the centroid-corrected average is independently on
the flow condition. Moreover, within the instrumental error, the local Pc − S
curve was the same as the average capillary pressure-saturation curve based on
the centroid-corrected average operator. During main drainage and main imbibition, the difference between each average phase pressure based on various
averaging operators was negligible.
88
Chapter 5. Local and average capillary pressure-saturation relationships
Bibliography
Hassanizadeh, S., O. Oung, and S. Manthay (2005), Laboratory experiments and
simulations on the significance of the non-equilibrium effect in capillary pressure saturation relationship, Unsaturated Soil: Experimental Studies , Proceedings of the International Conference From experimental evidence towards
numerical modeling of unsaturated soils, 93.
Helmig, R. (1997), Multiphase flow and transport processes in the subsurface: a
contribution to the modeling of hydrosystems., Springer.
Korteland, S., S. Bottero, S. Hassanizadeh, and C. Berentsen (2009), What is
the correct definition of average pressure?, Transport in Porous Media.
Liu, H., and H. Dane (1995), Improved computational procedure for retention
relations of ommiscible fluids using pressure cells, Soil Sci.Soc.Am., 59.
Nordbotten, J., M. Celia, H. Dahle, and S. Hassanizadeh (2007), Interpretation of macroscale variables in darcy’s law., Water Resources Research,
43 (W08430, doi:10.1029/2006WR005018).
Nordbotten, J., M. Celia, H. Dahle, and S. Hassanizadeh (2008), On the definition of macro-scale pressure for multi-phase flow in porous media., Water
Resources Research, 44 (W06502, doi:10.1029/2006WR005715).
Van Genuchten, M. (1980), A closed-form equation for predicting the hydraulic
conductivity of unsaturated soils, Soil Sci.Soc. Am. J., 44.
White, M., and M. Oostrom (1997), Stomp. subsurface transport over multiple
phases. user’s guide, Tech. rep., Pacific Northwest National Laboratory.
Chapter
6
Non-equilibrium two-phase
flow experiments1
6.1
Introduction
At the macro-scale, two-phase flow processes are usually modelled by the mass
balance equation and Darcy’s equation for each phase. When combined, these
result in the following Equation (Bear , 1972; Helmig, 1997).
φ
∂Sα
krα (Sα )
= ∇.
k.(∇Pα − ρα g) ;
∂t
µα
α = w, n
(6.1)
where subscripts w and n denotes wetting and nonwetting phases respectively.
All quantities appearing in these two equations are macro-scale quantities. Assuming that porosity, relative permeability and fluids properties are known, and
considering that Sw + Sn = 1 the unknowns in this set of equations are: Pn ,
Pw and Sw . This leads to three unknowns but only two equations. The closure
equation makes up for this deficit and states that the pressure difference between
the two phases is a function of wetting phase saturation:
Pn − Pw = f (Sw )
(6.2)
1 Prepared for submission to Advances in Water Resources, S.Bottero, S.M.Hassanizadeh,
P.J. Kleingeld
89
90
Chapter 6. Non-equilibrium two-phase flow experiments
This is also known as ’the capillary-pressure saturation relationship’. Because
in the literature, the fluids pressure difference is commonly assumed to be the
macroscale ’capillary pressure’ Pc Equation 6.2 is often rewritten as follows:
Pn − Pw = Pc (Sw )
(6.3)
This expression is a source of some of the misunderstanding associated with twophase flow theory. In fact, at the macro-scale, the pressure difference Pn − Pw is
not the same as capillary pressure Pc . The capillary pressure is a intrinsic property of the porous medium and two-fluids. For a meniscus capillary pressure is
defined, independently of the fluid pressures, by using Young-Laplace equation
as follows:
pc = σ
1
1
+
rx
ry
=
2σ
Rm
(6.4)
where σ represents the interfacial tension, rx and ry denote the principal radii of
curvature the meniscus, and Rm is the mean radius of curvature. Thus, through
rx and ry the capillary pressure depends on the pore dimension and through the
interfacial tension it depends on the surface properties of the fluids and the soil.
Equation 6.4 is valid whether the interface is moving or not. The relationship
between pn − pw and pc , however, depends on flow conditions. This relationship
can be derived from the force balance in the direction normal to the interface. In
this respect, Hassanizadeh and Gray (1993) derived the following force balance
equation for the direction N normal to an interface:
pn − pw =
2σ
+ N · (τn − τw ) · N − ∇σ · τwn · N
Rm
(6.5)
where pn and pw are microscale pressures of the two fluids on two sides of
the meniscus and τn and τw are their corresponding viscous stress tensors. It
is evident that in such a case pn − pw is not equal to pc . Even if interfacial
viscous forces are negligible or not present at all, pn − pw may be different form
pc due to dissipative forces within the fluids surrounding the meniscus. Indeed,
the significance of these forces effects has been shown for instance by Sheng and
Zhou (1992) who have studied the motion of a meniscus in a tube during piston
displacement of a wetting phase by a nonwetting phase or viceversa. They found
6.1. Introduction
91
pn − pw =
µq A
2σwn
±B
Rm
r
(6.6)
where pn and pw are averaged on the two side of the interface, µ is the
viscosity, q is the velocity, B and A are coefficients that control the velocitydependent ’capillary pressure’. Thus the relationship pn − pw = pc is valid at
the meniscus only under static condition. Under dynamic conditions, pn − pw
depends on the flow velocity, which at larger scales manifest itself as a chenage
in saturation with time. In principle, the relationship pn − pw = pc is valid at
the meniscus only under static condition. Even if one would have pn − pw = pc
al all menisci within an REV, under flow conditions, it does not mean that the
same can be said at macroscale. Because fluid pressures vary spatially within
each flowing phase, macroscale (or average) pressure values will be different from
pressure values at the interface. In fact, based on thermodynamic considerations,
Hassanizadeh and Gray (1990) and Kalaydjian (1992) suggested an equation
which relates the difference in the phase pressure Pn − Pw to a capillary pressure
Pc
Pn − Pw = Pc (Sw ) − τ (Sw )
∂Sw
∂t
(6.7)
where τ is a non-equilibrium coefficient. In this formulation, Pc is an intrinsic
property of the porous medium-fluids system, whereas fluids pressure difference
Pn − Pw is dependent on flow dynamics (and thus initial and boundary conditions). In the literature, the term ”static capillary pressure” (referring to Pc )
and ”dynamic capillary pressure” (referring to Pn − Pw ) has been used. This is a
potentially confusing terminology. In fact, there is only one capillary pressure, as
explained above. So, we propose to refer to Pn − Pw as phase pressure difference
or simply as pressure difference and Pc (Sw ) denotes the capillary pressure which
is measured experimentally as a function of saturation in quasi-static experiments. Obviously the pressure difference between the wetting and nonwetting
phase only under static condition is equal to a capillary pressure. Equation 6.7
suggests that at any given point in time, where equilibrium is disturbed, the
saturation will change to reestablish the equilibrium condition, and the coefficient τ controls this process. In this chapter, the behavior of the phase pressure
difference under non-equilibrium condition at the local scale is investigated. For
this purpose, a series of drainage experiments were carried out in a homogeneous sand column by applying large pressure steps to the inflowing fluid. The
experimental results from a non-equilibrium primary drainage experiments under different injection pressures are presented. The non-equilibrium coefficient
92
Chapter 6. Non-equilibrium two-phase flow experiments
is first calculated based on local pressures and saturation. Then, the τ coefficient was calculated based on average pressures and saturation. The results are
presented and discussed.
6.2
Overview of non-equilibrium experiments
In this section, a brief overview of the laboratory experiments used to investigate
macroscale nonequilibrium effects are presented. Most experiments reported in
the literature have concerned unsaturated flow (Topp et al., 1967; Smiles et al.,
1971; Vachaud et al., 1972; Stauffer , 1978; Wildenschild et al., 2001; Sakaki et al.,
2009). A comprehensive review of non-equilibrium laboratory experiments were
given by Hassanizadeh et al. (2002). Here the attention is limited to experiments
where two-phase flow was considered.
Lenhard et al. (1988) performed drainage experiments under transient conditions for three-phase flow (oil-water-air) in a homogeneous sand column. Water
and NAPL saturations were determined with a dual-energy gamma radiation
system and liquid pressures were measured using hydrophobic and hydrophillic
ceramic tensiometers. Pressure of the wetting phase and the nonwetting phase
were measured by means of five pair of tensiometers placed inside the sand at
five different elevations. A laboratory column was filled with sand on top of a
4-cm thick coarse gravel to a hight of 56.5 cm from the bottom. The sample
was initially fully saturated with water. Then a mixture of NAPL was ponded
on top of the water surface. Drainage of the water started after the outflow
elevation was positioned lower than the soil surface. Some of their results are
shown in Figure 6.22. In this chapter some of their findings are compared with
our experimental results.
Kalaydjian (1992) performed a series of imbibition experiments in a limestone
and sandstone samples with water and PCE as wetting and nonwetting fluids,
respectively. The pressures of the two phases were measured at four elevations
on the opposite sides of the sample using four pairs of pressure transducers.
The water saturation was measured by ultrasonic method that measured the
traveling time of an ultrasonic wave between two points. To obtain saturation
this method was calibrated with saturation measurement performed with a Xray CT-scanner. The experiments were performed at various flow rates. The
pressure difference was found to depend on the flow rate and it increased with
the flow rate. The rate of saturation decreased when the flow rate was increased.
Laboratory experiments where also carried out in Geodelft in a small sand
column with the intent to investigate the phase pressure difference-saturation
relationship. The experiments are described to some extent in Hassanizadeh
6.3. Non-equilibrium experiments
93
et al. (2005) and Manthey (2005). The experiments consisted of three primary
drainage and three main imbibition cycles. Drainage processes were performed
with an injection pressure of 16 kPa, 20 kPa, 25 kPa. PCE and water were
the nonwetting and the wetting fluids, respectively. The soil sample was 0.06
m in diameter and 0.03 m in height. It was initially saturated with water and
the nonwetting fluid was injected from above. A hydrophobic membrane was
used to prevent the flow of PCE out of the sample. Pressures of the two phases
were measured at the side of the sample at the center of the column height by
means of a couple of pore pressure transducers (Oung and Bezuijen, 2003). The
water saturation in the column was determined volumetrically, based on the
water pressure measurements in the outflow reservoir. Thus in contrast to the
pressure, the saturation was averaged over the whole column.
O’Carroll et al. (2005) performed a series of Multi-Steps Outflow experiments
(MSO) to explore whether agreement between observed and predicted MSO
could be improved through the inclusion of the so-called nonequilibrium term
(as in Equation 6.7) in the governing equations. The fluids considered were water
and PCE. The porous medium consisted of sand with a mean grain size of 0.026.
The setup was a column 9.6 cm high and 5.07 cm diameter. PCE was injected
from below. A nylon membrane on top of the soil in the column was used as
a capillary barrier for PCE allowing only water outflow. A teflon membrane
at the bottom was used as a capillary barrier for water. A second experiment
was performed without teflon membrane to investigate its effect on the outflow.
The pressure varied from 20 cm H2 O (1960 Pa) to 90 cm of H2 O (8820 Pa) for
step increments of 5 cm H2 O (500 Pa). The capillary pressure was determined
separately by means of a pressure cell system.
6.3
Non-equilibrium experiments
The experimental set-up used for these experiments was presented in Chapter
4. Primary drainage non-equilibrium experiments were carried out in a homogeneous sand column. The experiments began by applying a large pressure to the
inflowing fluid and keeping it constant throughout the experiment. The sample was initially fully saturated with water and PCE was injected from below.
Drainage experiments were carried out at different imposed pressures: 20 kPa,
30 kPa, 35 kPa, 38 kPa. The pressure of the wetting and nonwetting phases
as well as the saturation were measured inside the sand column at three different elevations z1 , z2 and z3 (see set-up in Figure 4.1). Local fluid pressures,
local water saturation, as well as the average saturation were recorded every
two seconds for fifteen hours, at constant temperature of 20◦ C. Water sensors
94
Chapter 6. Non-equilibrium two-phase flow experiments
were calibrated to measure hydrostatic pressure at the beginning of the primary
drainage whereas nonwetting phase pressure sensors were calibrated to measure
oilstatic pressure at the end of the drainage experiment when there was no flow.
6.3.1
Results from primary drainage experiments
Results from non-equilibrium primary drainage experiments for the four different
injection pressures are shown in Figures 6.1, 6.2, 6.3, 6.4. Figures 6.1 a, 6.2 a,
6.3 a, 6.4 a, show the wetting and nonwetting phase pressures as a function of
time for early times of the drainage process. A general pressure trend can be
noticed in all experiments. We can distinguish four main stages:
1. As soon as the inflow valve (see Figure 4.1) was opened, all pressure transducers registered a high value corresponding to the injection pressure.
2. The nonwetting phase flowed into the column displacing the wetting phase.
An initial dramatic pressure drop of both fluids is observed until the front
nonwetting phase pressure front arrives.
3. As soon as the PCE front reached a pair of transducers, a rapid increase of
pressure of both fluids was observed, with the pressure of the nonwetting
phase always higher than the pressure of the wetting phases. The front
reached the water and PCE sensors at the same time, indicating a relatively
homogeneous sand packing. This is an important requirement, as the local
phase pressure difference was determined by subtracting the pressure of the
nonwetting phase measured on one side of the column from the wetting
phase pressure measured on the opposite side. It should be mentioned
that no fingering effects were observed as the denser fluid was injected
from below and flowed upward.
4. After breakthrough of the nonwetting phase, the pressures slowly decreased.
This is more obvious for the water phase at later times when it tended to
return to hydrostatic equilibrium. For steady-state, the flow of the nonwetting phase continued.
Figures 6.1 c-d, show the change of water saturation in time for an injection
pressure of 20 kPa. As previously mentioned, the nonwetting phase is injected
from below, thus a lower water saturation is expected for the sensor at position
z1 that is closest to the bottom of the column. However, the water saturation
at this point was found to be higher than the water saturation measured at the
other sensors. This could be due to micro-heterogeneities in the sand packing or
preferential flow. This anomaly, however, was not observed in the other three
6.3. Non-equilibrium experiments
95
(a)
(b)
10
5
0
200
400
Pressure [kPa]
Pn1
Pn2
Pn3
Pn1
Pn2
Pn3
15
0
Water saturation [−]
20
10
5
time [s]
2
time [s]
(c)
(d)
1
1
0.8
0.8
0.6
0.4
Sw1
Sw2
Sw3
0.2
0
0
Pn1
Pn2
Pn3
Pw1
Pw2
Pw3
15
0
600
200
400
Water saturation [−]
Pressure [kPa]
20
1
0.2
0
1
8
8
6
4
Pn−Pw at z1
Pn−Pw at z2
Pn−Pw at z3
200
400
time [s]
2
time [s]
3
4
4
x 10
(f)
10
600
Pn−Pw, [kPa]
Pn−Pw, [kPa]
(e)
0
x 10
0.4
10
0
4
4
Sw1
Sw2
Sw3
time [s]
2
3
0.6
0
600
0
6
4
Pn−Pw at z1
Pn−Pw at z2
Pn−Pw at z3
2
0
0
1
2
time [s]
3
4
4
x 10
Figure 6.1: Non-equilibrium experiment results at imposed pressure of 20 kPa.(a)
and (b) wetting and nonwetting phase pressures versus early and later time respectively; (c) and (d) local water saturation versus time; (e) and (f ) local pressure difference versus time at early and later time.
96
Chapter 6. Non-equilibrium two-phase flow experiments
(a)
(b)
35
35
Pressure [kPa]
25
20
15
10
Pn1
Pn2
Pn3
Pw1
Pw2
Pw3
30
Pressure [kPa]
Pn1
Pn2
Pn3
Pw1
Pw2
Pw3
30
5
25
20
15
10
5
0
0
0
200
400
600
0
time [s]
(c)
0.6
0.4
0.2
0
200
400
Water saturation [−]
Sw1
Sw2
Sw3
0.6
0.4
0.2
time [s]
2000
4000
time [s]
(e)
(f)
10
10
8
8
6
4
Pn−Pw at z1
Pn−Pw at z2
Pn−Pw at z3
2
0
Sw1
Sw2
Sw3
0.8
0
600
200
400
time [s]
600
Pn−Pw, [kPa]
Water saturation [−]
Pn−Pw, [kPa]
1
0.8
0
6000
(d)
1
0
2000
4000
time [s]
0
6000
6
4
Pn−Pw at z1
Pn−Pw at z2
Pn−Pw at z3
2
0
0
2000
4000
time [s]
6000
Figure 6.2: Non-equilibrium experiments results at imposed pressure of 30 kPa.
(a) and (b) wetting and nonwetting phase pressures versus early and later time;
(c) and (d) local water saturation versus time; (e) and (f ) pressure difference
versus early and later time respectively.
6.3. Non-equilibrium experiments
97
(a)
(b)
40
Pn1
Pn2
Pn3
Pw1
Pw2
Pw3
30
20
10
Pressure [kPa]
Pressure [kPa]
40
0
Pn1
Pn2
Pn3
Pw1
Pw2
Pw3
30
20
10
0
0
200
400
600
0
time [s]
(c)
Sw1
Sw2
Sw3
0.6
0.4
0.2
0
200
400
Water saturation [−]
Water saturation [−]
1
0.8
0.6
0.4
0.2
0
600
Sw1
Sw2
Sw3
0.8
0
time [s]
8
8
6
4
Pc−Sw at z1
Pc−Sw at z2
Pc−Sw at z3
2
0
5000
time [s]
10000
(f)
10
200
400
time [s]
600
Pn−Pw, [kPa]
Pn−Pw, [kPa]
(e)
10
0
10000
(d)
1
0
5000
time [s]
6
4
Pn−Pw at z1
Pn−Pw at z2
Pn−Pw at z3
2
0
0
5000
time [s]
10000
Figure 6.3: Non-equilibrium experiment results at imposed pressure of 35 kPa.
(a) and (b) wetting and nonwetting phase pressures versus time; (c) and (d) local
water saturation versus time; (e) and (f ) local pressure difference versus time.
98
Chapter 6. Non-equilibrium two-phase flow experiments
(a)
(b)
40
Pn1
Pn2
Pn3
Pw1
Pw2
Pw3
30
20
10
Pressure [kPa]
Pressure [kPa]
40
0
Pn1
Pn2
Pn3
Pw1
Pw2
Pw3
30
20
10
0
0
500
time [s]
1000
0
(c)
Sw1
Sw2
Sw3
0.6
0.4
0.2
0
500
time [s]
Water saturation [−]
Water saturation [−]
1
0.8
0.6
0.4
0.2
0
1000
Sw1
Sw2
Sw3
0.8
0
(e)
8
8
Pn−Pw, [kPa]
Pn−Pw, [kPa]
10
6
4
Pn−Pw at z1
Pn−Pw at z2
Pn−Pw at z3
2
0
5000
time [s]
10000
(f)
10
0
10000
(d)
1
0
5000
time [s]
500
time [s]
1000
6
4
Pn−Pw at z1
Pn−Pw at z2
Pn−Pw at z3
2
0
0
5000
time [s]
10000
Figure 6.4: Non-equilibrium experiment results at imposed pressure of 38 kPa.
(a) and (b) wetting and non-wetting phase pressures versus time; (c) and (d)
local water saturation versus time; (e) and (f ) local capillary pressure.
6.4. The average non-equilibrium phase pressure difference
99
primary drainage experiments, whose results are shown in Figures 6.2 c-d, 6.3cd and 6.4 c-d. The change in water saturation was as expected, it decreased
rapidly as the PCE front reached a sensor and then leveled off as soon as it
passed. Plot of non-equilibrium pressure differences determined by subtracting
the water pressure from PCE pressure at each elevation are shown in Figures
6.1 e-f, 6.2 e-f, 6.3 e-f, 6.4 e-f. Except for the 20 kPa experiment, we see a
non-monotonous variation with time. There is an initial sharp increase of the
pressure difference to an overshoot and then decrease to an asymptotic value.
We note that this overshoot increases with the increase of the injecting pressure.
This overshoot was not observed in the case of 20 kPa (Figure 6.1 e-f).
Finally, for the four primary drainage experiments at injection pressures of 20
kPa, 30 kPa, 35 kPa and 38 kPa, graphs of non-equilibrium pressure difference
versus local water saturation are shown in Figures 6.5, 6.6, and 6.7 for the
three elevations z1 , z2 and z3 , respectively. In the same Figures is the local
capillary pressure-saturation curve also shown. The non-equilibrium pressure
difference curves lie significantly higher than the capillary pressure-saturation
curve. Also, the larger the injection pressure the great the difference between
the Pc − Sw curve and the pressure difference curve. For the case of primary
drainage performed at 20 kPa, the capillary pressure increased monotonically
with a decrease of water saturation. However, different behavior is observed
for experiments performed at high injection pressures. The experimental results
show that the non-equilibrium pressure difference has a non-monotonic trend
at all elevations. Moreover, for higher injection pressure a larger overshoot is
observed.
6.4
The average non-equilibrium phase pressure
difference
In this section, measured local phase pressures and local saturations are used
to calculate average phase pressure and average saturation representative of the
whole domain. The observation points are located at elevations z1 , z2 , z3 . An
extra observation point was taken as the bottom of the column, z0 . The nonwetting phase pressure at elevation z0 was assumed to be the same as the injection
pressure Pn0 = Pbott . It is also assumed that water saturation at this observation point reached residual saturation already from the very beginning of the
experiments. The residual saturation was assumed to be the same as the other
elevations at the end of each non-equilibrium experiment, and it can be directly
read off in Figures 6.5, 6.6 and 6.7. The local water pressure Pw0 at observation
100
Chapter 6. Non-equilibrium two-phase flow experiments
12
10
Pn−Pw, [kPa]
8
6
4
2
0
0
equilibrium
20 kPa
30 kPa
35 kPa
38 kPa
0.1
0.2
0.3
0.4
0.5
0.6
Water saturation [−]
0.7
0.8
0.9
1
Figure 6.5: Local non-equilibrium phase pressure difference-saturation curves at
applied pressure of 20 kPa, 30 kPa, 35 kPa, 38 kPa at elevation z1 .
12
10
Pn−Pw, [kPa]
8
6
4
2
0
0
equilibrium
20 kPa
30 kPa
35 kPa
38 kPa
0.1
0.2
0.3
0.4
0.5
0.6
Water saturation [−]
0.7
0.8
0.9
1
Figure 6.6: Local non-equilibrium phase pressure differences-saturation curves at
applied pressure of 20 kPa, 30 kPa, 35 kPa, 38 kPa at elevation z2 .
6.4. The average non-equilibrium phase pressure difference
101
12
10
Pn−Pw, [kPa]
8
6
4
2
0
0
equilibrium
20 kPa
30 kPa
33 kPa
35 kPa
0.1
0.2
0.3
0.4
0.5
0.6
Water saturation [−]
0.7
0.8
0.9
1
Figure 6.7: Local non-equilibrium phase pressure differences-saturation curves at
applied pressure of 20 kPa, 30 kPa, 35 kPa, 38 kPa at elevation z3 .
point z0 was determined by the difference between the non-equilibrium pressure
difference at residual saturation at elevation z1 , and the nonwetting pressure,
Pw0 = Pn0 − ∆P (Swr )|n.e . To eliminate noise, the raw data were smoothed by
using ′ rloess function’ from the Matlab library. Moreover, when the nonwetting
phase was not present its pressure was assumed to be the same as the wetting
phase. In Chapter 5 various pressure average operators were introduced. To
calculate the average phase pressures here two averaging operators were used:
intrinsic-phase average operator and the centroid-corrected average operator.
The average water saturation is determined arithmetically as shown in Equation
5.8. The water saturations and the phase pressures were averaged over the four
st st
st
observation points z0 , z1 , z2 , and z3 for four times t = tst
0 , t1 , t2 , and t3 when
the moving front reached each of them. The averaged variables at intermediate
time t = ti are then linearly interpolated. Once the nonwetting phase passed
the last sensor (i.e. once both phases were present in the averaging domain) the
averaging was performed for all measurements times. Results form the averaging procedure are shown in Figures 6.8, 6.9, 6.10, and 6.11. In these figures, the
non-equilibrium pressure differences determined by the intrinsic-phase operator and the centroid-corrected average operator are plotted versus the average
102
Chapter 6. Non-equilibrium two-phase flow experiments
20
18
16
<Pc>=<Pn>−<Pw> Centroind−corrected
<Pn>−<Pw>|n.e Intrinsic phase
<Pn>−<Pw>|n.e Centroid−corrected
Pn−Pw at z1
Pn−Pw at z2
Pn−Pw at z3
Pn − Pw, [kPa]
14
12
10
8
6
4
2
0
0
0.2
0.4
0.6
Water Saturation, Sw [−]
0.8
1
Figure 6.8: Average non-equilibrium pressure differences versus average wetting
saturation; corresponding local curve at three elevations z1 , z2 , and z3 for injection pressure of 20 kPa, and the average capillary pressure-saturation curve.
water saturation. In the same figure, the average capillary pressure is plotted
versus the corresponding average water saturation (this is described in Section
5.6.1). The local non-equilibrium pressure difference versus saturation is also
plotted at three elevations z1 , z2 and z3 . In Figure 6.8, the intrinsic-phase
average of non-equilibrium pressure difference lies above the average capillary
pressure. This is expected, as the injection pressure in case of non-equilibrium
experiment was much higher than the pressure applied for the equilibrium case.
By this operator, whenever there is an advancing front, high nonwetting phase
pressures are weighted by high nonwetting saturations. Also, the intrinsic-phase
average curve exhibits an overshoot that is not observed at the local scale. As
discussed in Chapter 3 and Chapter 5, this occurs because the centroids of the
two phases do not coincide with each other and with the centroid of the average
domain. The centroid-corrected average corrects the intrinsic phase average for
the distance between the centroid of a phase and the centroid of the domain.
Similar results are seen in Figures 6.9, 6.10, and 6.11, for injection pressure
of 30 kPa and 35 kPa and 38 kPa, respectively. For those cases the centroidcorrected pressure difference curves lie much higher than the average capillary
pressure. Moreover significantly, they show a non-monotonic behavior. This
6.5. The non-equilibrium coefficient τ at the local scale
103
25
Pn−Pw, [kPa]
20
<Pc>=<Pn>−<Pw> Centroid−corrected
<Pn>−<Pw>|n.e
Intrinsic phase
<Pn>−<Pw>|n.e
Centroid−corrected
Pn−Pw at z1
Pn−Pw at z2
Pn−Pw at z3
15
10
5
0
0
0.2
0.4
0.6
Water Saturation, Sw [−]
0.8
1
Figure 6.9: Average non-equilibrium pressure differences versus average wetting
saturation; corresponding local curve at three elevations z1 , z2 , and z3 for injection pressure of 30 kPa, and the average capillary pressure-saturation curve.
indicates that the non-equilibrium effect increases with the increase in injection
w
pressure (i.e. larger ∂S
∂t ). This effect is much more pronounced in the case
a of intrinsic-phase average. But, this stronger effect is considered to be an
artifact of intrinsic-phase average operator rather than the true non-equilibrium
effect. It should be noted that the centroid-corrected pressure difference curves
fall below the local scale pressure difference at higher saturation; (around 0.8
for the 20 kPa case and 0.9 for the 30 kPa case). This is perhaps an indicator
of the approximation nature of the centroid-corrected average operator at high
saturations (see Chapter 3).
6.5
The non-equilibrium coefficient τ at the local
scale
The experimental data presented in Figures 6.1, 6.2, 6.3, and 6.4, (in particular
plots of Pn − Pw versus time and Sw versus time), can be used to determine
the non-equilibrium coefficient τ by employing Equation 6.7. According to this
equation, τ is the slope of the curve when (Pn − Pw ) − Pc is plotted versus
104
Chapter 6. Non-equilibrium two-phase flow experiments
25
Pn−Pw, [kPa]
20
<Pc>=<Pn>−<Pw> Centroid−corrected
<Pn>−<Pw>|n.e Intrinsic phase
<Pn>−<Pw>|n.e Centroid−corrected
Pn−Pw at z1
Pn−Pw at z2
Pn−Pw at z3
15
10
5
0
0
0.2
0.4
0.6
Water Saturation, Sw [−]
0.8
1
Figure 6.10: Average non-equilibrium pressure differences versus average wetting saturation; corresponding local curve at three elevations z1 , z2 , and z3 for
injection pressure of 35 kPa, and the average capillary pressure-saturation curve.
∂Sw /∂t. Such a plot must be made for a specific saturation and for a given
measurement point. This will result in finding τ as a function of saturation. The
procedure is illustrated for the measurement point z1 . First, for each injection
pressure, plots of (Pn − Pw ) − Pc versus Sw (Figure 6.12 a) and ∂Sw /∂t versus
Sw ( Figure 6.12 b) were made. From Sw − t curves, ∂Sw /∂t was calculated
using a central difference scheme. To reduce the noise, the time derivative was
smoothed by the robust locally weighted scatter smooth method which uses non
locally-weighted linear regression to smooth the data (see MatLab function).
Next, at a given saturation Sw , values of (Pn − Pw ) − Pc and ∂Sw /∂t were read
and plotted against each other, as shown in Figure 6.13. At a given saturation,
this procedure provides four data points, one for each injection pressure. We also
assumed that if ∂Sw /∂t = 0, then there should be no difference between capillary
pressure and pressure difference (i.e. (Pn − Pw ) − Pc = 0 at ∂Sw /∂t = 0). This
resulted in a fifth data point. These five points were interpolated by a straight
line as shown in Figure 6.13. It must be noted that in this fitting we forced the
line through the origin.
The slope of the resulting line is an estimate of τ at the corresponding saturation. The data points were fitted with regression curve and optimized applying
6.5. The non-equilibrium coefficient τ at the local scale
105
25
<Pc>=<Pn>−<Pw> Centroid−corrected
<Pn>−<Pw>|n.eq Intrinsic−phase
<Pn>−<Pw>|n.eq Corrected−centroid
Pc−Sw at z1
Pc−Sw at z2
Pc−Sw at z3
<Pn>−<Pw>, [kPa]
20
15
10
5
0
0
0.2
0.4
0.6
Water Saturation, Sw [−]
0.8
1
Figure 6.11: Average non-equilibrium pressure differences versus average wetting saturation; corresponding local curve at three elevations z1 , z2 , and z3 for
injection pressure of 38 kPa, and the average capillary pressure-saturation curve.
(a)
(b)
7
0.005
20 kPa
30 kPa
35 kPa
38 kPa
6
0
−0.005
−0.01
DSw/Dt [1/s]
Pn−Pw−Pc, [kPa]
5
20 kPa
30 kPa
35 kPa
38 kPa
4
3
−0.015
−0.02
−0.025
−0.03
2
−0.035
1
−0.04
0
0.4
0.5
0.6
0.7
Water Saturation [−]
0.8
−0.045
0.4
0.6
0.8
Water Saturation [−]
1
Figure 6.12: a) (Pn − Pw ) − Pc versus water saturation; b) ∂Sw /∂t versus water
saturation at elevation z1 for injection pressure 20 kPa, 30 kPa, 35 kPa and 38
kPa.
106
Chapter 6. Non-equilibrium two-phase flow experiments
6000
6000
5000
5000
4000
4000
Pn−Pw−Pc [Pa]
Pn−Pw−Pc [Pa]
a least-square Marquardt-Levenberg algorithm, with ∂Sw /∂t as the independent
variable and (Pn − Pw ) − Pc as dependent variable. Table 6.1 shows τ value with
a 95 % percent of confidence bounds and the degree of goodness (R2 ).
3000
fit Sw=85
Sw=0.85
fit Sw=80
Sw=0.80
fit Sw=75
Sw=0.75
fit Sw=70
Sw=0.70
2000
1000
0
−0.04
3000
fit Sw=0.65
Sw=0.65
fit Sw=0.60
Sw=0.60
fit Sw=0.55
Sw=0.55
fit Sw=0.50
Sw=0.50
2000
1000
−0.03
−0.02
dSw/dt [1/s]
−0.01
0
0
−0.04
−0.03
−0.02
dSw/dt [1/s]
−0.01
0
Figure 6.13: (Pn − Pw ) − Pc versus ∂Sw /∂t at various water saturation. The
slope of each curve represent the material coefficient τ .
Finally, in Figure 6.14 the non-equilibrium coefficient τ was plotted versus
water saturation. It decreases in the range 0.75 < Sw < 0.85 and increases in
the range 0.50 < Sw < 0.75. It can be seen that the magnitude of this coefficient
at the local scale varies between 1.70 · 105 (Pa.s)to 1.35 · 105 (Pa.s).
An alternative fitting procedure is not to force the regression though the
origin. This implies that more than one coefficient may be needed to describe
the non-equilibrium effect. In other words, Equation 6.7 is considered to hold
over a range of ∂Sw /∂t away from equilibrium point ∂Sw /∂t = 0:
(Pn − Pw ) − Pc = −τ
∂Sw
+I
∂t
(6.8)
where I represents the intercept. Such graphs and the resulting regression lines
are shown in Figure 6.15. In Table 6.2, the values of tau are reported including
the 95 % of confidence bounds, the error and the goodness of the fitting. A
similar procedure was used by Dahle et al. (2005) and Manthey (2005). It must
be noted that in all cases, the value of the intercept is much smaller than the
6.5. The non-equilibrium coefficient τ at the local scale
107
Table 6.1: The non-equilibrium coefficient τ estimated at the local scale versus
local water saturation Sw .
Sw
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
τ [P a.s]
1.59 · 105
1.45 · 105
1.36 · 105
1.37 · 105
1.40 · 105
1.40 · 105
1.46 · 105
1.69 · 105
95 % confidence bounds
1.88 · 105 , 1.29 · 105
1.66 · 105 , 1.24 · 105
1.53 · 105 , 1.19 · 105
1.51 · 105 , 1.24 · 105
1.66 · 105 , 1.15 · 105
1.78 · 105 , 1.02 · 105
2.08 · 105 , 0.84 · 105
2.52 · 105 , 0.86 · 105
R2
0.94
0.96
0.97
0.98
0.93
0.96
0.93
0.93
5
1.7
x 10
1.65
tau [Pa.s]
1.6
1.55
1.5
1.45
1.4
1.35
0.5
0.55
0.6
0.65
0.7
0.75
Water saturation, Sw [−]
0.8
0.85
Figure 6.14: Non-equilibrium coefficient τ at local scale versus water saturation.
(Pn − Pw ) − Pc .
Figure 6.16 shows τ as function of saturation. By using Equation 6.8 the
non-equilibrium coefficient follows a different trend compared with the method
shown in Figure 6.14. In fact it decreases with a decrease in water saturation in
the range of saturation considered. However, as shown in Table 6.2 the degree
of confidence decreases at lower saturation. A third method of estimating τ ,
adopted by Kalaydjian (1992) is to consider each measurement obtained in each
Chapter 6. Non-equilibrium two-phase flow experiments
6000
6000
5000
5000
4000
4000
(Pn−Pw)−Pc, [Pa]
(Pn−Pw)−Pc, [Pa]
108
3000
3000
Sw=0.85
fit Sw=0.85
Sw=0.80
fit Sw=0.80
Sw=0.75
fit Sw=0.75
Sw=0.70
fit Sw=0.70
2000
1000
0
−0.04
2000
1000
−0.02
dSw/dt, [1/s]
0
0
Sw= 0.65
fit Sw= 0.65
Sw= 0.60
fit Sw= 0.60
Sw= 0.55
fit Sw= 0.55
Sw= 0.50
fit Sw= 0.50
−0.04
−0.02
dSw/dt, [1/s]
0
Figure 6.15: (Pn − Pw ) − Pc versus ∂Sw /∂t at various water saturation. The
slope of each curve represents the material coefficient τ . The regression curve is
not forced through the origin as in Equation 6.8.
5
1.5
x 10
1.45
1.4
tau, [Pa.s]
1.35
1.3
1.25
1.2
1.15
1.1
0.5
0.55
0.6
0.65
0.7
0.75
Water saturation, Sw [−]
0.8
0.85
Figure 6.16: The coefficient τ versus water saturation according to Equation 6.8.
6.6. The non-equilibrium coefficient τ at the column scale
109
Table 6.2: Values of the non-equilibrium coefficient τ at the local scale for different water saturations. Regression curve not forced through origin, (Pn − Pw ) −
w
Pc = −τ ∂S
∂t + I.
Sw
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
τ [P a.s]
1.49 · 105
1.37 · 105
1.28 · 105
1.29 · 105
1.25 · 105
1.17 · 105
1.10 · 105
1.21 · 105
95 % bounds
2.16 · 105 , 0.82 · 105
1.84 · 105 , 0.90 · 105
1.63 · 105 , 0.92 · 105
1.55 · 105 , 1.03 · 105
1.69 · 105 , 0.79 · 105
1.81 · 105 , 0.53 · 105
2.06 · 105 , 0.13 · 105
2.45 · 105 , 0.03 · 105
I
240
210
240
230
430
620
900
990
95% bounds
1.696, -1.21
1.299, -0.88
1.096, -0.62
0.842, -0.39
1.444, -0.58
2.007, -0.76
2.778, -0.97
2.961, -0.97
R2
0.94
0.97
0.98
0.99
0.96
0.92
0.81
0.76
experiment, independent of other measurements. In this cases at a given water
saturation, ∂Sw /∂t and Pn −Pw are substituted into Equation 6.7. The resulting
non-equilibrium coefficient is plotted versus water saturation in Figure 6.17, for
various injection pressure. For the low injection pressure of 20 kPa, τ increases
with a decrease in water saturation and it is higher than the coefficient calculated
for pressures of 30 kPa, 35 kPa and 38 kPa. However, for high injection pressures
this coefficient does not depend on the applied pressure. Also, the trend and the
magnitude are similar to the coefficient calculated by the first method, (see
Figure 6.14).
6.6
The non-equilibrium coefficient τ at the column scale
In the previous section, the procedure to estimate the dynamic coefficient based
on the phase pressures and saturation at local scale was presented. The same
methodology can be used to estimate the non-equilibrium coefficient at the column scale. At the column scale the non-equilibrium coefficient is expressed as:
< τ >= −
< Pn > − < Pw > − < Pc >
∂
∂t < Sw >
(6.9)
Thus, at any given average saturation for each injection pressure, there is a
pair of < Pn > − < Pw > − < P c > and ∂ < Sw > /∂t values. Non-equilibrium
average pressure differences in comparison with the average capillary pressure
are shown in Figures 6.8, 6.9, 6.10 and 6.11. We denote < τ >i as the nonequilibrium coefficient based on the intrinsic-phase average pressures, and [τ ]
110
Chapter 6. Non-equilibrium two-phase flow experiments
5
4
x 10
20 kPa
30 kPa
35 kPa
38 kPa
3.5
tau, [Pa.s]
3
2.5
2
1.5
1
0.5
0.55
0.6
0.65
0.7
0.75
Water saturation, Sw [−]
0.8
0.85
Figure 6.17: The non-equilibrium coefficient τ versus local water saturation for
different injection pressures, by method three.
10
<Pn>−<Pw>|eq
Fit van Genughten
9
8
<Pn>−<Pw>, [kPa]
7
6
5
4
3
2
1
0
0
0.2
0.4
0.6
Average water saturation, <Sw> [−]
0.8
1
Figure 6.18: Fitted average capillary pressure curve by van Genughten model.
6.6. The non-equilibrium coefficient τ at the column scale
111
as the non-equilibrium coefficient based on the centroid-corrected average phase
pressures. For the calculation of these coefficients, first the average capillary
pressure is fitted with the Van Genuchten model. The resulting curve is shown
in Figure 6.18 and the corresponding van Genuchten parameter values are given
in Table 6.18. The derivative with respect to time of the average saturation was
calculated by a centered difference scheme.
Table 6.3: Van Genuchten parameters for average capillary pressure.
Average equilibrium
8.713
0.0002079
0.99
8000
8000
7000
7000
6000
6000
Pn−Pw−Pc, [Pa]
Pn−Pw−Pc, [Pa]
VG parameters
n
α
R2
5000
4000
Sw = 85
fit 85
Sw = 80
fit 80
Sw = 75
fit 75
Sw = 70
fit 70
3000
2000
1000
0
−5
−4
−3
−2
dSw/dt, [1/s]
5000
4000
3000
2000
1000
0
−4
−1
−3
x 10
Sw=65
fit 65
Sw = 60
fit 60
Sw =55
fit 55
Sw = 50
fit 50
−3
−2
dSw/dt, [1/s]
−1
−3
x 10
Figure 6.19: Difference between the average phase pressure difference, [Pn ]−[Pw ],
and the capillary pressure versus the rate of change of the average water saturation, ∂ < Sw > /∂t. In the same figure the regression curves for various water
saturation are plotted. The slope of each curve represents the non-equilibrium
coefficient, [τ ] at the column scale.
In Figure 6.19, the difference between the average non-equilibrium pressure
difference, based on the centroid-corrected averaging operator, ([Pn ] − [Pw ]),
and average capillary pressure is plotted versus ∂ < Sw > /∂t for various average water saturation values (< Sw >= 0.50, .., 0.85). At each saturation, the
112
Chapter 6. Non-equilibrium two-phase flow experiments
4
4
x 10
1.8
1.6
1.6
1.4
1.4
<Pn>−<Pw>−Pc, [Pa]
<Pn>−<Pw>−Pc, [Pa]
1.8
1.2
1
Sw=85
fit 85
Sw=80
fit 80
Sw=75
fit 75
Sw=70
fit 70
0.8
0.6
0.4
0.2
−5
−4
−3
−2
dSw/dt, [1/s]
x 10
1.2
1
0.8
0.6
0.4
0.2
−4
−1
−3
x 10
Sw=65
fit 65
Sw=60
fit 60
Sw=55
fit 55
Sw=50
fit 50
−3
−2
dSw/dt, [1/s]
−1
−3
x 10
Figure 6.20: Difference between the average phase pressure difference < Pn >i
− < Pw >i and the capillary pressure versus the rate of change of the average
water saturation. In the same figure the regression curves for various water
saturation are plotted. The slope of each curve represents the non-equilibrium
coefficient at the column scale < τ >i .
6.6. The non-equilibrium coefficient τ at the column scale
113
6
x 10
Centroid−corrected
Intrinsic phase
6
<tau>, [Pa.s]
5
4
3
2
1
0
0.5
0.55
0.6
0.65
0.7
0.75
Average water saturation, <Sw> [−]
0.8
0.85
Figure 6.21: Average non-equilibrium coefficients [τ ] and < τ > based on the
centroid-corrected and intrinsic phase average pressure respectively.
four points are fitted by a regression curve. The slope of the regression curve
represents the average non-equilibrium coefficient. In this case, the origin of
coordinate axes (0;0) is not included in the regression. In Table 6.4, the average
values of the non-equilibrium coefficient [τ ] are reported, as well as the 95 %
of bounds confidence interval and R2 . In Figure 6.21 the non-equilibrium coefficient based on the centroid-corrected average pressures [τ ], is plotted versus
water saturation. It is significant to see that for the entire saturation range,
the average non-equilibrium coefficient is one order of magnitude larger than
the non-equilibrium coefficient a the local scale, (see Table 6.1 and Figure 6.14).
This can be also seen from the magnitude of the phase pressure difference and
rate change of saturation. In fact at the average scale the rate change of saturation is much smaller that the local one. Moreover the difference between the
average pressure difference and the average capillary pressure is larger at the
column scale than at the local scale. In Figure 6.20, the difference of the average non-equilibrium pressure difference, < Pn >i − < Pw >i and the average
capillary pressure is plotted versus the rate change of average saturation for a
range of water saturations (Sw = 0.5, .., 0.85). The values of the non-equilibrium
coefficient < τ >i are reported in Table 6.5. In Figure 6.21, the coefficient
114
Chapter 6. Non-equilibrium two-phase flow experiments
< τ >i is plotted versus average water saturation. It is clear that the value of
the coefficient < τ >i for the entire range of water saturations is larger that the
values obtained based on the centroid-corrected, [τ ].
Table 6.4: Non-equilibrium coefficient [τ ] at the column scale estimated at various
average water saturation < Sw > .
[τ ], (P a.s)
1.09 · 106
8.25 · 105
1.41 · 106
1.38 · 106
1.44 · 106
2.40 · 106
1.95 · 106
1.50 · 106
< Sw >
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
95 % bounds
(4.33 · 106 , −2.14 · 106 )
(3.24 · 106 , −1.59 · 106 )
(1.99 · 106 , 8.24 · 105 )
(1.49 · 106 , 1.26 · 106 )
(1.66 · 106 , 1.23 · 106 )
(3.08 · 106 , 1.72 · 106 )
(3.09 · 106 , 8.06 · 105 )
(3.29 · 106 , −2.93 · 105 )
I
372.4
1905
1138
1435
1220
156
313
566
95 % bounds
(1.21 · 104 , −1.14 · 104 )
(1.06 · 104 , −6808)
(2921, −644.8)
(1788, 1081)
(5188, 61.3)
(161, −129)
(2760, −213)
(4406 − 327)
R2
0.51
0.52
0.98
0.99
0.99
0.99
0.96
0.86
Table 6.5: Non-equilibrium coefficient < τ >i at the column scale estimated at
various average water saturation < Sw >.
< Sw >
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
6.7
< τ >i , (P a.s)
1.65 · 106
1.71 · 106
2.87 · 106
3.01 · 106
3.27 · 106
5.92 · 106
5.89 · 106
5.58 · 106
95 % bounds
(4.74 · 106 , −1.43 · 106 )
(4.55 · 106 , −1.12 · 106 )
(3.72 · 106 , 2.012 · 106 )
(4.15 · 106 , 1.87 · 106 )
(4.77 · 106 , 1.78 · 106 )
(6.88 · 106 , 4.96 · 106 )
(7.27 · 106 , 4.50 · 106 )
(7.75 · 106 , 3.40 · 106 )
I
7183
7484
4880
4004
2704
748
2170
2991
95% bounds
(1.84 · 104 , 4068)
(1.77 · 104 , 2752)
(7477, 2284)
(7468, 539)
(7252, 1843)
(2794, 1298)
(5131, 791)
(7639, -1657)
R2
0.73
0.77
0.99
0.98
0.98
0.98
0.99
0.98
Discussions
Figures 6.5, 6.6 and 6.7 showed that throughout the primary drainage experiments, the non-equilibrium pressure difference-saturation curves lie above the
capillary pressure-saturation curve. Moreover, a non-monotonic behavior is seen
for high injection pressure. In the literature, there are only few laboratory experiments the reported such non-monotonic behavior, (see Hassanizadeh et al.,
2005; Lenhard et al., 1988). This is probably because there are only a few nonequilibrium experiments where both fluid pressures are measured at serial points
along the column. In almost all of the two-phase flow experiments, fluid pressures are measured in the fluids just outside the column. Also, saturation is
determined by volumetric or gravimetric methods from the outflow, that gives
6.7. Discussions
115
column-scale average values. Obviously, in dynamic experiments, the pressure
measured outside the sample does not represent the pressure inside it.
Figure 6.22: Measured versus predicted water and oil pressure head from. The
solid symbol represents the water phase while the open symbol the oil phase pressure. The continuous and dashed lines represent the results of the numerical
simulations Lenhard et al., 1988.
Figure 6.23: Measured water saturation versus time from Lenhard et al., 1988.
Lenhard et al. (1988) performed a drainage experiment under non-equilibrium
condition to investigate three-phase flow (oil-water-air) processes in a porous
116
Chapter 6. Non-equilibrium two-phase flow experiments
medium. Although this experiment involved to a more complex system, some of
their results can be compared with the results shown in this thesis. Figure 6.22
shows graphs of a measured and predicted water and oil pressure heads at 25.2
cm depth. The solid symbols represent the water phase while the open symbol
the oil phase pressure. The continuous and dashed lines represent results of the
numerical simulations that used two different methods. Both pressures show a
non-monotonic behavior with a peak around 50 minutes. This obviously yield
to a non-monotonic behavior of the head pressure difference, hn − hw , versus
time in the first 100 min. In Figure 6.23, it is also observed that in this time
frame the water saturation decreases from 1 to 0.4. The comparison between
the predicted value and the measured phase pressures are not in good agreement
in the first part of the drainage processes. The predicted curves did not reproduced the experimental data. Pressure difference overshoot was also found by
Hassanizadeh et al. (2005) in drainage experiments on a small sand column, 6 cm
in diameter and 3 cm high. The fluids pressures were measured by transducers
installed at the side of the column, while the average saturation was determined from the volume of the outflow. Berentsen et al. (2005) simulated these
drainage experiments. It was found that the pressure phase difference overshoot
was reproducible only by including the non-equilibrium term in the governing
equations as suggested in Equation 6.7. Estimation of the non-equilibrium coefficient τ based on our experimental results shows that this coefficient is a function
of water saturation. However, the nature of the relationship between the nonequilibrium coefficient and saturation is not clear. Different trends are obtained
depending on the method used to estimate it. Assuming a linear relation ∂Sw /∂t
and (Pn − Pw ) − Pc , τ shows a ’parabolic shape’. The non-equilibrium coefficient
first decreases with decrease in water saturation for the range 0.85 > Sw > 0.75
and then increases between 0.75 > Sw > 0.50. Different trend is observed by
assuming an intercept as suggested by Dahle et al. (2005) and Manthey (2005).
In this case τ decreases with the decrease of saturation. This relationship τ − Sw
is in discrepancy to what was found by O’Carroll et al. (2005) and Berentsen
et al. (2005). To simulated their experimental results, they employed a function
of τ (Se ) which increased with a decrease in the effective saturation. Similar
results are also shown in Chapter 7 where the experimental results are compared to a simulations. An open question is whether the value of the coefficient
τ varies with length scale. The results presented here show that this coefficient is scale dependent. In fact, it increases by one order of magnitude from
the local scale to the column scale when centroid-corrected averaging operator
is employed. At the local scale, it is found that τ varies from a maximum of
1.58 · 105 (Pa.s) to a minimum of 1.36 · 105 (Pa.s). While at the column scale
6.7. Discussions
117
it varies between 2.3 · 106 (Pa.s) and 1.39 · 106 (Pa.s). Obviously larger values
are obtained when the intrinsic-phase operator is used. It should be noted, that
this coefficient does not scale with the length square as suggested by Dahle et al.
(2005). Hassanizadeh et al. (2005) and Manthey (2005) found for the same type
of porous medium that τ varied from 1.44 · 105 (Pa.s) to 0.65 · 105 (Pa.s) for
water saturations between 0.70 and 0.90. This coefficient can be refered to as
local value. In fact, the phase pressure was measured by a sensor having a diameter of 0.7 cm, which was almost three times smaller than the total length of
column. However, the water saturation was not measured inside the sand but
was obtained from outflow fluxes. Thus, the water saturation represented an
average quantity over the whole sand sample. This has obvious consequences on
the estimation of the coefficient τ . The absolute value of the rate of change of
the average saturation is smaller that the time derivative of the local saturation.
Nevertheless, their results are comparable with the non-equilibrium coefficient
obtained in our non-equilibrium drainage experiment at the sensor scale or ’local
scale’. O’Carroll et al. (2005) found from Multistep outflow (MSO) experiments
that a better match between experimental and numerical results was obtained
when including the non-equilibrium term in the governing equations describing
two-phase flow processes. In their case, τ was not calculated directly from measurements but it was estimated from a parameter optimization algorithm. They
found a satisfactory fit when a linear relation for τ versus effective saturation
was adopted (τ = −A · Se + A, where A was a constant). Thus, τ increased
with a decrease in effective water saturation. As already mentioned in the section 6.2 regarding the overview on experimental work, they performed two MSO
experiments which differed from each other by the presence of a hydrophobic
membrane at the bottom of the column. The material coefficient τ was estimated for both experiments. The maximum value found for the coefficient τ
was 5.64 · 107 (Pa.s) for the experiments without membrane, and 1.99 · 107 (Pa.s)
for the experiments with membrane. These values are local values in the sense
that τ is assigned to each node inside of the model domain. In order to compare our results with O’Carroll’s results, we calculated this coefficient at a water
saturation of 0.8 based on the linear function they used. At this saturation,
τ = 1.1 · 107 (Pa.s) for the experiment without membrane and τ = 3.9 · 106
(Pa.s) with membrane. These values are much higher that the values obtained
at the local scale in our experiments. Stauffer (1978) suggested an empirical
expression for the non-equilibrium coefficient τ . It reads:
αµw φ
τS =
Kλ
Pd
ρw g
2
(6.10)
118
Chapter 6. Non-equilibrium two-phase flow experiments
According to this formula, a larger coefficient should be determined for a fine
sand with a low intrinsic permeability and a high entry pressure. The intrinsic
permeability in O’Carroll’s experiment was one order larger than the one considered in our experiment. Thus, τ should be smaller than what was found in
our experiment. For the same experiments O’Carroll et al. (2005) also calculated < τ > that is representative of the whole domain. The average pressure
difference was calculated by subtracting the water pressure measured at the top
reservoir from the nonwetting phase pressure imposed at the bottom of the column. The rate of change of saturation was calculated based on the average
water saturation at the end of the outflow step at an effective saturation of 0.50.
They found a non-equilibrium coefficient τ = 2.82 · 107 (Pa.s) comparable to the
local value (τ = 1.63 · 107 Pa.s). Thus, according to their procedure, there is
no dependence of τ on the length of the domain. However, it should be noted
that what they called ’average pressure difference’ determined from the pressure
difference of the fluid reservoirs does not necessarily represent the average pressure difference. The phase pressures at the boundaries are not the same as the
average phase pressures calculated from local pressure by an average operator.
Manthey et al. (2008) defined a dynamic number given by the ratio of dynamic
forces and viscous forces as follow:
Dy =
kτ
Dynamic capillary f orces
=
µw lc2 φ
viscous f orces
(6.11)
where k is the scalar intrinsic permeability, µw is viscosity of the wetting phase,
lc is the characteristic length and φ is porosity. The definition of a characteristics
length is not clear and; obviously its choice would yield different results. The
characteristic length could be the size of the front, the length of the domain, or
the size of the pore depending on which length scale the processes are measured.
We employed this number to estimate the importance of dynamic forces with
respect to the viscous forces at the sensor scale (or local scale) and the column
scale. In Table 6.6, the various parameters used for this calculation are listed.
Intrinsic permeability and porosity are assumed to be constant along the column.
The dynamic coefficients are estimated for water saturation Sw = 0.7. The
characteristic lengths are the diameter of the sensor lc = 7 · 10−3 m and the
length of the averaging domain < Lc >≃ 0.18 m. Based on the choice of this set
of parameters the dynamic number at larger scale is Dy= 0.30, indicating that
at this scale the viscous forces prevail. At the sensor scale the dynamic number
is found to be two orders of magnitude larger, Dy= 26.6, suggesting that the
dynamic capillary forces are dominant with respect the viscous forces. Thus, the
importance of the dynamic forces diminishes with an increase in the length scale
6.8. Summary and conclusions
119
Table 6.6: Dynamic number calculate at the local and column scale.
Parameters
k
φ
µw
lc
Lc
τ
<τ >
Dy
<Dy>
value
2.8 · 10−12
0.4
1 · 10−3
7 · 10−3
0.18
1.37 · 105
1.39 · 106
26.6
0.30
unit
m2
[-]
Pa.s
m
m
Pa.s
Pa.s
[-]
[-]
of observation.
6.8
Summary and conclusions
A series of primary drainage experiments was carried out to investigate nonequilibrium effects in the fluid pressure difference-saturation relationship. The
experimental results showed that throughout the non-equilibrium process, the
pressure difference-saturation curves lied higher than the capillary pressuresaturation curve. Moreover, the non-equilibrium pressure difference showed
a non-monotonic behavior with an overshoot which was more pronounced at
higher injection pressures. Based on local measured pressures and saturation,
average pressures fluid difference and saturation were calculated for a column
domain. Two average operators were employed: the intrinsic-phase average and
the centroid-corrected average. As it was already shown for the equilibrium case,
the intrinsic average operator overestimates the non-wetting phase average. This
averaging effect is quite pronounced in the non-equilibrium case as the pressures
applied are much larger. It was found that the average non-equilibrium pressure difference based on the centroid-corrected average operator still lied above
the average capillary pressure and at large injection pressures it presents a nonmonotonic behavior. The non-equilibrium coefficient τ was calculated at the
local scale by three different methods. Although each method showed different
trend of τ on respect to saturation, its value was found to be of the order of 105
(Pa.s). The non-equilibrium coefficient at the column scale was also estimated
based on average pressure and saturation. The results show that this coefficient is one order of magnitude higher than the local non-equilibrium coefficient,
around 106 (Pa.s). This suggests that the dynamic coefficient is scale length
dependent.
120
Chapter 6. Non-equilibrium two-phase flow experiments
Bibliography
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Berentsen, C., S. Hassanizadeh, and O. Bezuijen, A.and Oung (2005), Modelling
of two-phase flow in porous media including non-equilibrium capillary pressure
effects, Proceeding Computational Methods in Water Resounces.
Dahle, H., M. Celia, and S. Hassanizadeh (2005), Bundle-of tubes model for
calculating dynamic effects in the capillary-pressure-saturation relationship,
Transport in Porous Media, 58.
Hassanizadeh, S., and W. Gray (1990), Mechanics and thermodynamics of multiphase flow in porous media, Advances Water Resources, 13.
Hassanizadeh, S., and W. Gray (1993), Thermodinamic basis of capillary pressurein porous media., Water Resources Research., 29.
Hassanizadeh, S., M. Celia, and H. Dahle (2002), Dynamic effect in the capillary
pressure-saturation relationship and its impact on unsaturated flow, Vadose
Zone Hydrology, 1.
Hassanizadeh, S., O. Oung, and S. Manthay (2005), Laboratory experiments and
simulations on the significance of the non-equilibrium effect in capillary pressure saturation relationship, Unsaturated Soil: Experimental Studies , Proceedings of the International Conference From experimental evidence towards
numerical modeling of unsaturated soils, 93.
Helmig, R. (1997), Multiphase flow and transport processes in the subsurface: a
contribution to the modeling of hydrosystems., Springer.
Kalaydjian, F. (1992), Dynamic capillary pressure curve for water/oil displacement in porous media, theory vs. experiments, Proc. SPE Conference, Washington DC. SPE, Brookfield, CT.
Lenhard, R., J. Dane, J. Parker, and J. Kaluarachchi (1988), Measurrements
and simulation of one-dimentional transient three-phase flow for monotonic
liquid drainage, Water Resources Research, 24 (6), 853–863.
Manthey, S. (2005), Two-phase flow processes with dynamic effect in porous
media-parameter estimation and simulation, Ph.D. thesis, Stuttgart University.
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Manthey, S., S. Hassanizadeh, R. Helmig, and R. Hilfer (2008), Dimensional
analysis of two-phase flow including a rate-dependent capillary pressuresaturation relationship, Advances in Water Resources, 31 (9), 1137–1150.
O’Carroll, D., T. Phelan, and L. Abriola (2005), Exploring dynamic effect in capillary pressure in multistep outflow experiments, Water Resources Research,
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in geotechnics, 4.
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water retention curves: Direct laboratory quantification of dynamic coefficient
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Soil Sci.Soc. Amer. Proc., 35.
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Chapter
7
Numerical simulation of
non-equilibrium two-phase
experiments1
7.1
Description of the numerical code
A series of numerical simulations were carried out to simulate the laboratory
experiments described in Chapter 6. Two kinds of simulations were performed.
The first type was based on the traditional governing two-phase flow equations.
In the second type, the non-equilibrium term suggested by Hassanizadeh and
Gray (1993a,b) was included. The numerical model was developed by Berentsen
et al. (2005). Here, for the sake of completeness the model is briefly described.
Two-phase flow processes are described by the mass balance equations and
Darcy’s law for each flow phase. Assuming incompressible fluids in the absence
of any sink or source, the combination of the mass balance and Darcy’s law lead
to the following flow equations:
∂Sw
kkrw
φ
+∇·
∇Φw = 0
(7.1)
∂t
µw
∂Sn
kkrn
φ
+∇·
∇Φn = 0
(7.2)
∂t
µn
1 Prepared for submission to Water Resources Research, S.Bottero, S.M.Hassanizadeh,
C.W.J. Berentsen
123
124
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
where the subscript w and n indicates the wetting and the nonwetting phases
respectively, φ is the porosity, S is saturation, k is the intrinsic permeability
tensor [L2 ], kr is the phase relative permeability, µ is the dynamic viscosity
[M L−1 T −1 ], Φ is the phase potential [M T −2 L−1 ]. The phase potentials are
expressed as follows:
Φw = Pw + ρw gz;
Φn = Pn + ρn gz;
(7.3)
where P is the pressure [M T −2 L−1 ], ρ is density [M L−3 ], g is gravity acceleration [LT −2 ]. The governing equations contain six unknowns, (Φn ), (Φw ), Sn ,
Sw krn , krw . Thus additional equations are needed. Equation 7.4 describes the
saturation of one fluid phase as a function of the other fluid phase:
Sw + Sn = 1
(7.4)
Another equation relates the relative permeability of a phase with saturation
kr (Se ). The Burdine model (Burdine, 1953) is adopted:
2+3λ
λ
krw (Se ) =
(7.5)
Se
2
krn (Se ) =
(1 − Se ) (1 − Se )
2+λ
λ
(7.6)
The effective saturation Se is defined as:
Se =
Sw − Srn
1 − Srn − Srw
(7.7)
Srn is the residual nonwetting phase saturation and Srw is the irreducible wetting
phase saturation. The closure equation is defined as follows:
Φn − Φw − (ρn − ρw )gz = Pc (Sw ) − τ
∂Sw
∂t
(7.8)
The Brooks-Corey formulation is used to describe the Pc (Se ) constitutive relationship:
Pc (Se ) = Pd Se−1/λ
f or
Pc > Pd
(7.9)
Pc (Se ) = Pd
f or
Pc < Pd
(7.10)
7.2. Conceptual model
125
where Pd is the entry pressure, and λ indicate the pore size distribution, σ
and µ are fitting coefficients. In this numerical study, various functions that
related the non-equilibrium coefficient τ to the wetting saturation are used, as
listed below:
τ (Sw ) =
τ0
τ (Sw ) =
τ0 [Sw ]
τ (Sw ) =
τ (Sw ) =
τ (Sw ) =
(7.11)
(7.12)
2
τ0 [(Sw ) ]
(7.13)
2
2
(7.14)
2
(7.15)
τ0 [exp(−Sw − µ) /(2σ )]
τ0 [(1 + erf ([Sw − µ]/σ ))]
The set of Equations 7.1, 7.2, and 7.8 were solved numerically by a finite elements
technique. They are discretized in space by a first-order upwind scheme. In this
way, only the values upwind or upstream from the control volume are used
to calculate the value of the parameter inside the grid cell. For the temporal
discretization, a fully implicit scheme was applied.
7.2
Conceptual model
Figure 7.1: Conceptual model
126
7.2.1
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
Domain description and medium-fluids properties
The sketch of the model domain is presented in Figure 7.1. The domain has
the same length as the column in the laboratory experiments. It was discretized
into uniform cells of size of d =10−4 m. The optimum dimension of a cell was
established from preliminary non-equilibrium numerical experiments where the
size was reduced until no variation in the solution occurred. The domain was
divided into sub-domains consisting of the sand, the holder and filters. The fluids
used in experiments were water and PCE whose properties at 20◦ C are listed in
Table 7.1 as well as the porosity and the intrinsic permeability of the filters and
holder. The hydraulic conductivity of the brass filter was determined by using
the falling head method. The hydraulic conductivity of the holder was determined
based on a harmonic average after measuring the hydraulic conductivity of the
brass filter and holder together.
Lb
Lh
Lbh
=
+
Kbh
Kb
Kh
(7.16)
The porosity of the brass filter was specified by the manufacturer. The holder’s
porosity was determined volumetrically. The hydraulic conductivity of the sand
was determined prior to each experiment by a constant head method. This test
was repeated four times and the arithmetic average was used. Because fluid
pressures were measured in situ at three elevations, the hydraulic conductivity
was also determined locally. In some cases, a slight difference in the hydraulic
conductivity between two observation points were observed. These were due heterogeneities in the sand which are taken into account to properly reproduce the
laboratory experiments. Thus, we distinguish four sand layers: ’sand1 ’, ’sand2 ’,
’sand3 ’, and ’sand4 ’ as shown in Figure 7.1. The local porosity of the sand
was determined by using the time domain reflectometry (TDR) knowing that
at the beginning of each drainage experiment only the water phase was present.
The error in the water saturation measurement was of the 2% percent. A maximum absolute error of 5% percent was encountered in the hydraulic conductivity.
Thus, preliminary simulations were performed to determine the intrinsic permeability and porosity values by varying them until the best fit to all measured data
were obtained. This was done for each experiment and the results are reported
in Table 7.2. The Brooks-Corey parameters used as input in the numerical simulations were determined by using a parameter optimization algorithm (trust
region, see MatLab library) from the local capillary pressure-saturation curves
obtained in equilibrium experiments (Figure 7.2). The capillary pressure Pc
was chosen as an independent variable and Sw as dependent variable. In this
7.2. Conceptual model
127
optimization, the points having Pc = 0 were excluded. The entry pressure of
the brass filter for water-PCE was determined experimentally by increasing the
pressure at the bottom of the empty column and allowing PCE to flow into the
column upward. The brass filter was first de-aired in a vacuum. The holder’s
entry pressure was assumed to be 100 Pa, which is much smaller than the entry
pressure of the brass filter and the sand. This is quite reasonable considering
the large dimensions of the radial and circular channels in the holder compared
to the voids in the sand. The irreducible water saturation of the brass filter was
assumed to be the same as the sand. The irreducible saturation of the holder
was assumed to be zero.
Table 7.1: Fluids and medium properties at 20◦ C, and Brooks-Corey parameters.
Fluids property
density water, ρw
density PCE, ρn
viscosity water, µw
viscosity PCE, µn
Intrinsic permeability brass filter
porosity φ brass filter
Intrinsic permeability holder
porosity φ holder
Brooks-Corey parameters
Sand Pd
λ coefficient sand
Sand Srw
Filter Pd
Filter Srw
Holder Pd
7.2.2
value
1000
1623
1 · 10−3
0.9 · 10−3
3e−12
0.40
3e−10
0.30
values
4060
5.045
0.097
2000
0.097
100
unit
kgm−3
kgm−3
Pa s
Pa s
m2
[-]
m2
[-]
unit
Pa
[-]
[-]
Pa
[-]
Pa
Boundary and initial conditions
The domain is initially fully water saturated thus:
SnInitial = 0
The total flow was zero, thus:
kkn
kkw
∇
∇Φn + ∇
∇Φw = 0
µn
µw
(7.17)
(7.18)
128
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
Table 7.2: Intrinsic permeability k, and porosity φ of four different zones in the
column between measurement levels z1 , z2 , z3 .
Medium property
k1 , [m2 ]
k2 , [m2 ]
k3 , [m2 ]
k4 , [m2 ]
φ1 ,[-]
φ2 ,[-]
φ3 ,[-]
φ4 ,[-]
20 kPa
2.4 · 10−12
2.7 · 10−12
2.7 · 10−12
2.4 · 10−12
0.39
0.405
0.40
0.395
30 kPa
2.7 · 10−12
2.7 · 10−12
2.7 · 10−12
2.7 · 10−12
0.39
0.39
0.39
0.39
35 kPa
2.7 · 10−12
2.8 · 10−12
2.8 · 10−12
2.8 · 10−12
0.39
0.40
0.40
0.40
38 kPa
2.64 · 10−12
2.64 · 10−12
2.64 · 10−12
2.64 · 10−12
0.40
0.40
0.40
0.40
The top and bottom boundary conditions were specified at virtual boundary
layer with properties listed in Table 7.3. The permeability of these thin layers
were assumed to be higher than the intrinsic permeability of the sand; this means
that their resistance to the flow was negligible.
Table 7.3: Properties of the top and bottom virtual boundary layers.
properties
k bot
k top
bot
Swr
top
Swr
bot
Srn
top
Srn
PCE saturation initial,Snbot
PCE saturation initial,Sntop
values
3 · 10−10
3 · 10−10
0
0
0
0
1
0
unit
[m2 ]
[m2 ]
[-]
[-]
[-]
[-]
[-]
[-]
The nonwetting phase’s bottom boundary pressure was specified to be:
Pnbot = Pninj + ρn gH
(7.19)
where Pninj is the pressure applied in the experiment. The bottom boundary
condition assigned to the wetting phase is:
Pwbot = Pn − Pdbot
(7.20)
In the experiments, the top boundary was exposed to atmospheric pressure.
7.3. Averaging simulated fluid pressures
129
10000
fitted BC
Pc1−Sw1
Pc2−Sw2
Pc3−Sw3
9000
Capillary pressure [Pa]
8000
7000
6000
5000
4000
3000
2000
1000
0
0
0.2
0.4
0.6
Water saturation [−]
0.8
1
Figure 7.2: Local capillary pressure-saturation fitted by Brooks-Corey model.
Thus:
Pn ≃ Pw
(7.21)
At the side boundaries, no-flow conditions were assigned qα = 0, α = w, n.
7.3
Averaging simulated fluid pressures
The solution of flow equations subject to initial and boundary conditions presented above resulted in the fields of saturation and pressure, calculated at given
grid cell. However, the measurement windows of pressure transducers and TDR
sensors, were much larger than a grid cell. Therefore, to make a proper comparison of the numerical and experimental results, grid cell values were averaged
over sizes domains corresponding to measurement window size. As we already
discussed in Chapter 5, while averaging saturation is straight forward, it is not
clear what is the correct way of averaging pressure. It was shown that the intrinsic phase average operator might not be the correct manner of averaging phase
pressure. In fact, even under static condition it overestimates the non-wetting
phase pressure, which results in an overestimation on the capillary pressure. Another averaging operator, the centroid-corrected average, was then introduced
130
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
10
IPA at 20 kPa
CCA at 20 kPa
IPA at 30 kPa
CCA at 30 kPa
IPA at 35 kPa
CCA at 35 kPa
9
8
<pn>−<pw> [kPa]
7
6
5
4
3
2
1
0
0.1
0.2
0.3
0.4
0.5
0.6
time [sec]
0.7
0.8
0.9
1
Figure 7.3: Simulated average pressure differences using intrinsic-phase averaging and centroid-corrected averaging operators. The average is performed over
an area of 1 cm in diameter.
which corrected the intrinsic phase average for the difference between the centroid of the α-phase and the centroid of the averaging domain. The output of
a pressure transducer is also an average. In our case, it is the average of a fluid
pressure along a surface in contact with the sand. However, how a sensor averages pressure is not known. Under non-equilibrium conditions the employed
averaging operator may play an important role even at the sensor scale. Here,
the effect on the pressure difference that the intrinsic phase average or the centroid corrected averaging operators has at the sensor scale is shown. Consider
the simulations of non-equilibrium experiments with bottom boundary pressures
of 20 kPa, 30 kPa, 35 kPa. Let’s average the phase pressure along an averaging
window of one cm in diameter at the middle of the column. Note that these
simulations were performed without implementing the non-equilibrium capillarity term in the constitutive equations. The centroid-corrected average pressure
is calculated based on Equation 5.12. The centroid of the domain is for this
specific case at the middle of the circular surface of the pressure sensor in contact with the sand. In Equations 5.15 and 5.16 the subscripts ’bot ’ and ’top’
represent the first and the last nodes of the averaging domain, in the z-direction,
respectively. Figure 7.3 shows the simulated pressure difference versus water
7.4. Results and discussion
131
3
tau=0 at 20 kPa
tau=0 at 30 kPa
tau=0, 35 kPa
(<Pn>i−<Pw>i)− ([Pn]−[Pw]), [kPa]
2.5
2
1.5
1
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
time [s]
0.7
0.8
0.9
1
Figure 7.4: Differences in results obtained by intrinsic-phase averaging and the
centroid-corrected averaging operators.
saturation at the sensor scale at three lower boundary pressures. The dashed
line represents the intrinsic phase average and the continuous line the centroidcorrected averaging operator. In all simulations, the centroid-corrected lies below
the intrinsic phase average. Moreover, the difference between the two averages
increases with an increase in the boundary pressure. Figure 7.4 shows the difference between the pressure difference calculated by the intrinsic-phase averaging
and the pressure difference determined by the centroid-corrected averaging operators, (< Pn >i − < Pw >i ) − ([Pn ] − [Pw ]), versus average water saturation.
Large differences occur at high water saturations and at high boundary pressures. Thus, under non-equilibrium conditions, the average operator used to
obtain the average pressure plays a significant role even at the sensor scale. In
the following sections, the centroid-corrected average operator is employed to
compare with the experimental data.
7.4
Results and discussion
Two series of numerical simulations were performed to reproduce the experimental data. The first series were carried out by using the traditional macroscopic
132
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
equations and the equilibrium capillary pressure-saturation relationship. The
second series of simulations were performed by introducing the non-equilibrium
term as in Equation 7.8. For the dependence of the τ coefficient on the saturation Sw , the functions listed in Table 7.4 were used. In this section, results of
each simulation are compared with the experimental data and discussed.
7.4.1
Simulation of drainage experiments at 20 kPa
Table 7.4: The non-equilibrium coefficient τ0
Simulation
linear
quadratic
error function
exponential
τ [P a.s] 20 kPa
0.5 · 105
0.7 · 105
0.7 · 105
0.7 · 105
30 kPa
1 · 105
2 · 105
2 · 105
2 · 105
35 kPa
1 · 105
2 · 105
2 · 105
2 · 105
38 kPa
2 · 105
5 · 105
5 · 105
5 · 105
Figure 7.5(a) shows a comparison between the measured and simulated phase
pressures versus time at position z1 for an injection pressure of 20 kPa. Here
the attention will be focused on the first 600 s of the drainage experiment. In
this time frame, most of the wetting phase is displaced by the nonwetting phase;
thus, the non-equilibrium effect is more pronounced. The simulations did not
produce the scatter observed in the measurements. Data scattering of ±250P a
are already present in the injection pressure data. This effect was caused by the
on/off switching of the pumps used on the recirculation system. This scatter is
not taken into account for the bottom boundary pressure in the simulations. In
the simulations, the bottom pressure is kept constant. The simulation without
the dynamic term, τ0 = 0, reproduced the general trend of both wetting and
nonwetting phase pressures. Also, note that before the nonwetting front reaches
the observation point z1 at 85 sec, no simulation data points are shown for the
nonwetting pressure because the manner of averaging. Both centroid-corrected
and intrinsic-phase averaging operators do not yield an average value for pressure if the phase is not present in the averaging domain. This is not the case for
the water phase as it is always presents from the start of the measurement. As
soon as the nonwetting phase reaches the first node in z1 the nonwetting phase
average pressure is defined. Also in Figure 7.5, results of simulations including
the non-equilibrium capillary theory (employing functional forms for τ (Se ) are
plotted. The τ0 values adopted for each function are listed in Table 7.4. Simulated fluid pressures are plotted in Figure 7.5(a). Figure 7.5(c) shows measured
and simulated pressure differences versus time obtained with the standard twophase flow equations as well as for non-equilibrium. It is evident that all models
7.4. Results and discussion
133
20
nonwetting phase
pressure [kPa]
15
10
wetting phase
5
0
0
100
200
300
time [s]
400
500
600
1
data
tau=0
tau=0.7e+5 err−func
tau=0.7e+5 quadratic
tau=0.7e+5 exponential
tau=0.5e+5 linear
Water saturation [−]
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
100
200
300
time [s]
400
500
600
0
100
200
300
time[s]
400
500
600
10
Pn−Pw, [kPa]
8
6
4
2
0
Figure 7.5: Comparison of simulated and measured curves versus time at elevation z1 for the injection pressure of 20 kPa: a) phase pressures; b) water
saturation; c) non-equilibrium pressure difference.
134
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
20
nonwetting phase
pressure [kPa]
15
10
5
wetting phase
0
0
100
200
300
time [sec]
400
500
600
1
Water saturation [−]
0.9
0.8
0.7
0.6
data
tau=0
tau=0.7e+5 err
tau=0.7e+5 quadr
tau=0.7e+5 exp
tau=0.5e+5 linear
0.5
0.4
0.3
0.2
0
100
200
300
time [s]
400
500
600
0
100
200
300
time[s]
400
500
600
10
Pn−Pw [kPa]
8
6
4
2
0
Figure 7.6: Comparison of simulated and measured curves versus time at elevation z2 for the injection pressure of 20 kPa a) phase pressures; b) water
saturation; c) non-equilibrium pressure difference.
7.4. Results and discussion
135
20
pressure [kPa]
15
nonwetting wetting phase
10
5
0
0
100
200
300
time [s]
400
500
600
1
Water saturation [−]
0.9
0.8
data
tau=0
tau=0.7e+5 err
tau=0.7e+5 quadratic
tau=0.7e+5 exponential
tau=0.5e+5 linear
0.7
0.6
0.5
0.4
0.3
0.2
0
100
200
300
time [s]
400
500
600
0
100
200
300
time[s]
400
500
600
10
Pn−Pw, [kPa]
8
6
4
2
0
Figure 7.7: Comparison of simulated and measured curves versus time at elevation z3 for the injection pressure of 20 kPa a) phase pressures; b) water
saturation; c) non-equilibrium pressure difference.
136
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
follow the general trend of the measured pressure difference relatively well. The
simulated pressure difference increases monotonically as observed in the measurement. The simulated pressure difference lies below the measured one. When
the dynamic term is included in the model, a slight increase in the pressure difference is observed at the first stage of the front development. However, it is
clear that for small values of τ , there is not much difference between the simulated curves with and without the dynamic coefficient. Figures 7.5 (b) shows
the local water saturation versus time at z1 . The simulation with τ = 0 shows
an early arrival time for the nonwetting front. As soon the front reaches the
observation point z1 and moves towards the end of the domain, a large change
in water saturation from 1 to 0.4 is calculated. This is not in agreement with the
measurements. At breakthrough 50% of water still remains in the column. By
employing the dynamic term, the results does not improve. In fact, the dynamic
term yields a larger water displacement. A difference of 10% in water saturation
cannot be justified by the error in measurements (2%). The usual suspect for
causing this poor agreement is the presence of heterogeneities. A better agreement could be obtained by optimizing the irreducible water saturation. Similar
results were obtained for measurements at location z2 and z3 in Figures 7.6 (a),
(b), (c) and Figures 7.7 (a), (b), (c) respectively. Figure 7.8 shows the measured
and simulated column-scale average non-wetting phase saturation versus time.
It should be reminded that the average PCE saturation was measured at the
outflow of the column and it includes not only the volume of PCE in the sand
column but also the PCE volume inside the holder and the filters. These are
taken into account in the simulations. The comparison shows good agreement
with the simulated results for both with and without the non-equilibrium term
(τ = 0) and measured curves.
7.4.2
Simulation of drainage experiments at 30 kPa
Figure 7.9 (a) shows the measured and simulated wetting and nonwetting pressures versus time at observation point z1 . Let’s first consider the case when
τ = 0, i.e without the non-equilibrium term. A steep increase in both phase pressures occurred as soon the nonwetting front reached z1 . The manner the simulated nonwetting phase pressure increases to approach a relatively constant pressure (after breakthrough) differs from the measurements. Before breakthrough
(around 300 s) the simulation underestimate the measured nonwetting phase
pressure. An improvement is observed when τ (Se ) is included in the governing
equations. The non-equilibrium term leads to an increase in the nonwetting
phase pressure compared to the case with τ = 0. Figure 7.9 (c) shows the measured pressure difference versus time along with the simulated curves obtained
7.4. Results and discussion
137
1.2
Average PCE saturation [s]
1
0.8
data
tau=0
tau=0.7e+5 err
tau=0.7e+5 quadratic
tau=0.7e+5 exponential
tau=0.5e+5 linear
0.6
0.4
0.2
0
−0.2
0
200
400
600
800
1000
time [s]
Figure 7.8: Measured and simulated column-scale average non-wetting phase saturation versus time at injection pressure of 20 kPa.
with and without the nonequilibrium model. The measured pressure difference
shows a distinct overshoot. It is clearly shown that this non-monotonic behavior
cannot be reproduced by the equilibrium model (τ = 0). In this case, the pressure difference increases monotonically with time. The non-monotonic pressure
behavior, however, is reproduced by the non-equilibrium model. This general
behavior can be reproduced by all of the functions employed here (see Equations
7.11). However, after the pressure difference reached its maximum peak , all the
curves decreased much faster that what was observed in the measurement. This
suggests that the functions chosen here are either adequate only within a certain
range of saturation or that the non-equilibrium equation may not describe the
entire process, i.e. extra terms may be needed. After breakthrough, all simulated pressure difference curves lie close to each other. This suggests that the
non-equilibrium term plays a significant role only when the front moves along
the observation points z1 . The non-equilibrium term affected not only the phase
pressure distribution but it also influenced the local water saturation distribution. Figure 7.9 (b) shows the measured and simulated wetting phase saturation
versus time. The non-equilibrium term shows an improvement compared to the
τ = 0 case, but an earlier and steeper decrease in saturation compared to the
138
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
30
nonwettng phase
pressure [kPa]
25
20
15
10
wetting phase
0
100
200
300
time [s]
400
500
600
Water saturation [−]
0.9
0.8
data
tau=0
tau=2e5, errfunction
tau=2e5, quadratic
tau=2e5, exponential
tau=1e5, linear
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
100
200
300
time [s]
400
500
600
0
100
200
300
time[s]
400
500
600
10
Pn−Pw, [kPa]
8
6
4
2
0
Figure 7.9: Comparison of simulated and measured curves versus time at elevation z1 for the injection pressure of 30 kPa a) phase pressures; b) water
saturation; c) non-equilibrium pressure difference
7.4. Results and discussion
139
30
pressure [kPa]
25
nonwetting
20
15
10
wetting
5
0
0
100
200
300
time [s]
500
600
data
tau=0
tau=2e5, errfunction
tau=2e5, quadratic
tau=2e5, exponential
tau=1e5, linear
0.9
Water saturation [−]
400
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
100
200
300
time [s]
400
500
600
0
100
200
300
time[s]
400
500
600
10
Pn−Pw, [kPa]
8
6
4
2
0
Figure 7.10: Comparison of simulated and measured curves versus time at elevation z2 for the injection pressure of 30 kPa a) phase pressures; b) water
saturation; c) non-equilibrium pressure difference
140
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
20
nonwetting
pressure [kPa]
15
10
5
0
wetting
0
100
200
300
time [s]
500
600
data
tau=0
tau=2e5, errfunction
tau=2e5, quadratic
tau=2e5, exponential
tau=1e5, linear
0.9
Water saturation [−]
400
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
100
200
300
time [s]
400
500
600
0
100
200
300
time[s]
400
500
600
10
Pn−Pw, [kPa]
8
6
4
2
0
Figure 7.11: Comparison of simulated and measured curves versus time at elevation z3 for the injection pressure of 30 kPa a) phase pressures; b) water
saturation; c) non-equilibrium pressure difference
7.4. Results and discussion
141
1
0.9
0.8
Average PCE saturation [−]
0.7
0.6
0.5
0.4
0.3
data
tau=0
tau=2e5, errfunction
tau=2e5, quadratic
tau=2e5, exponential
tau=1e5, linear
0.2
0.1
0
0
100
200
300
time [s]
400
500
600
Figure 7.12: Measured and simulated average PCE saturation versus time at
injection pressure of 30 kPa.
data is observed. Similar results were obtained for measurement point at z2 and
z3 , see Figures 7.10 and 7.11, respectively. Figure 7.12 shows the column-scale
average nonwetting phase saturation (including filters and holder) versus time.
Prior to breakthrough, the non-equilibrium term yields a steeper increase in the
average nonwetting phase saturation. This is in agreement with the measurements. However, the simulated profile shows a sharp flattening of the slope at
around 270 s while the measurement shows a gradual change to a higher steadystate value. In all of the simulations, except for the linear function τ (Se ), the
maximum value of the coefficient τ was 2 · 105 P a.s. For the linear case, a value
of 1 · 105 was employed. The values used in these simulations are larger than the
τ0 values employed for simulation at an injection pressure of 20 kPa.
7.4.3
Simulation of drainage experiments at 35 kPa
Figures 7.13, 7.14, and 7.15 show curves of measured and simulated phase pressures, water saturation, and pressure difference versus time at elevations z1 ,
z2 ,and z3 , respectively. Similar behaviors to those observed in the primary
drainage at 30 kPa are seen. Figure 7.13 (a) shows the wetting and nonwetting phase average pressures versus time at z1 . As soon as the nonwetting front
142
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
35
pressure [kPa]
30
25
20
15
nonwetting phase
10
wetting phase
5
0
0
100
200
300
time [s]
500
600
data
tau=0
tau=2e5, exponential
tau=2e5, errfunction
tau=2e5, quadratic
0.9
Water saturation [−]
400
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
100
200
300
time [s]
400
500
600
0
100
200
300
time[s]
400
500
600
10
Pn−Pw, [kPa]
8
6
4
2
0
Figure 7.13: Comparison of simulated and measured curves versus time at elevation z1 for the injection pressure of 35 kPa: a) phase pressures; b) water
saturation; c) non-equilibrium pressure difference.
7.4. Results and discussion
143
30
pressure [kPa]
25
20
15
10
nonwetting phase
wetting phase
5
0
0
100
200
300
time [sec]
500
600
data
tau=0
tau=2e5, exponential
tau=2e5, errfunction
tau=2e5, quadratic
0.9
Water saturation [−]
400
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
100
200
300
time [s]
400
500
600
0
100
200
300
time[s]
400
500
600
10
Pn−Pw, [kPa]
8
6
4
2
0
Figure 7.14: Comparison of simulated and measured curves versus time at elevation z2 for the injection pressure of 35 kPa: a) phase pressures; b) water
saturation; c) non-equilibrium pressure difference.
144
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
30
nonwetting phase
25
pressure [kPa]
wetting phase
20
15
10
5
0
0
100
200
300
time [sec]
500
600
data
tau=0
tau=2e5, exponential
tau=2e5, errfunction
tau=2e5, quadratic
0.9
Water saturation [−]
400
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
100
200
300
time [s]
400
500
600
0
100
200
300
time[s]
400
500
600
10
Pn−Pw, [kPa]
8
6
4
2
0
Figure 7.15: Comparison of simulated and measured curves versus time at elevation z3 for the injection pressure of 35 kPa: a) phase pressures; b) water
saturation; c) non-equilibrium pressure difference.
7.4. Results and discussion
145
0.8
0.7
Average PCE saturation [−]
0.6
0.5
0.4
0.3
0.2
data
tau=0
tau=0.2*1e6, exponential
tau=0.2*1e6, errfunction
tau=0.2*1e6, quadratic
tau=0.2*5e5, linear
0.1
0
0
100
200
300
time [s]
400
500
600
Figure 7.16: Measured and simulated average PCE saturation versus time at
injection pressure of 35 kPa.
reached the first node in z1 , both phase pressures increased sharply. However, for
the nonwetting phase pressure, this increase is more pronounced. Figure 7.13
(c) shows the non-equilibrium pressure difference versus time. It can be seen
that the overshoot observed in the measurements is reproduced only by adding
the non-equilibrium term to the governing equations. With τ = 0, the pressure difference increases monotonically towards a constant value after the front
reaches z1 . The non-monotonic behavior can be reproduced by using any of the
functions τ (Se ) suggested in Table 7.4. Here, also after the maximum peak is
reached, the simulated pressure difference decreased faster than the measured
curve. The simulated curves underestimated the measured values. Increasing
τ0 yielded a better reproduction of the overshoot, i.e. in the first 200 sec when
the front moved along the column. However, it did not improve the result after
the front passed z1 . The simulated pressure difference curves with and without
τ , lie close to each other. This discrepancy between measured and simulated
curves might suggests that either the functions adopted here are valid only for
certain range of saturation, and/or that more terms are needed in the closure
equation. Figure 7.13 (b) shows the measured and simulated water saturation
versus time. A better match to the saturation profile for 1 > Sw > 0.4 is ob-
146
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
served when the dynamic term is included. Moreover, the dynamic term yields
a larger displacement of the nonwetting phase. This is in agreement with the
measurements. Figure 7.16 shows measured and simulated average nonwetting
phase saturation versus time. Here again all simulated curves show a sharp kink
which is not observe in the data.
7.4.4
Simulation of drainage experiments at 38 kPa
Figure 7.17 (a) shows the simulated and measured fluid pressures versus time at
elevation z1 for the case of injection pressure of 38 kPa. Similar to the cases of
30 kPa and 35 kPa, the general trend in pressure could be reproduced only if the
non-equilibrium term was introduced. However, in the first 200 s the simulated
nonwetting phase pressure underestimated the measured values. Also in this
case the non-monotonic behavior of the pressure difference could be reproduced
only by employing the non-equilibrium term. This behavior can be reproduced
by any of the functional relationship τ (Sw ). The value of τ0 employed for linear
function τ (Se ) is 2 · 105 Pa.s. For the other functions, a τ0 of 5 · 105 Pa.s was
used. This value is higher than the τ0 values employed to simulate experiments
at lower injection pressures boundary. This was also at the observation points
z2 and z3 (see Figures 7.18 and 7.19).
7.5
Summary and Conclusions
Non-equilibrium primary drainage laboratory experiments were simulated first
by employing the traditional two-phase flow equations and then including the
dynamic term as suggested by Kalaydjian (1992) and Hassanizadeh and Gray
(1990). Various functional dependencies of the non-equilibrium coefficient τ on
wetting phase saturation namely linear, quadratic, error-function and exponential were considered. To compare simulated results with experimental data, the
nonwetting and wetting phase pressures as well as wetting saturation were averaged over an area comparable with the measurement window of the pressure
transducers and TDR sensor (about 1cm in diameter). Under non-equilibrium
conditions, the averaging operator chosen to average the phase pressure plays
an important role even at this small scale. In fact depending on the boundary pressure, large differences in the average pressures can be obtained whether
the intrinsic-phase averaging operator or the centroid-corrected operator is used.
The centroid-corrected average was employed to compare experimental data with
numerical results. Results show that for the injection pressure of 20 kPa, nonequilibrium effects on fluid pressure differences were not significant. The general
7.5. Summary and Conclusions
147
40
pressure [kPa]
35
30
25
20
nonwetting phase
wetting phase
15
10
0
100
200
300
time [sec]
500
600
data
tau=5e5 quadratic
tau=5e5 err−function
tau=5e5 exponential
tau=2e5 linear
tau=0
0.9
Water saturation [−]
400
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
100
200
300
time [s]
400
500
600
0
100
200
300
time[s]
400
500
600
10
Pn−Pw, [kPa]
8
6
4
2
0
Figure 7.17: Comparison of simulated and measured curves versus time at elevation z1 for the injection pressure of 38 kPa: a) phase pressures; b) water
saturation; c) non-equilibrium pressure difference.
148
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
30
pressure [kPa]
25
20
15
10
nonwetting phase wetting phase
5
0
0
100
200
300
time [s]
400
600
data
tau=5e5 quadratic
tau=5e5 err−function
tau=5e5 exponential
tau=2e5 linear
tau=0
0.9
Water saturation [−]
500
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
100
200
300
time [s]
400
500
600
0
100
200
300
time[s]
400
500
600
10
Pn−Pw, [kPa]
8
6
4
2
0
Figure 7.18: Comparison of simulated and measured curves versus time at elevation z2 for the injection pressure of 38 kPa: a) phase pressures; b) water
saturation; c) non-equilibrium pressure difference.
7.5. Summary and Conclusions
149
30
pressure [kPa]
25
nonwetting phase
20
15
10
5
wetting phase
0
0
100
200
300
time [s]
400
500
600
1
data
tau=5e5 quadratic
tau=5e5 err−function
tau=5e5 exponential
tau=2e5 linear
tau=0
Water saturation [−]
0.8
0.6
0.4
0.2
0
0
100
200
300
time [s]
400
500
600
0
100
200
300
time[s]
400
500
600
10
Pn−Pw, [kPa]
8
6
4
2
0
Figure 7.19: Comparison of simulated and measured curves versus time at elevation z3 for the injection pressure of 38 kPa: a) phase pressures; b) water
saturation; c) non-equilibrium pressure difference.
150
Chapter 7. Numerical simulation of non-equilibrium two-phase experiments
trend could be reproduce by the traditional equations as well as by including
the non-equilibrium term. The experimental data at high injection pressures
of 30 kPa, 35 kPa and 38 kPa exhibited an overshoot in the curve of nonequilibrium pressure difference versus time. Numerical results revealed that this
non-monotonic behavior can be reproduced only if the non-equilibrium term is included in the equations describing two-phase flow. Although the non-equilibrium
term improved the results and reproduced the general trends, a satisfactory fit to
the experimental data was not obtained. After the maximum peak of the overshoot was reached, the simulated pressure difference decreased more rapidly that
the observed one. Discrepancy between observed and simulated pressure difference curves might caused by the functional form chosen for τ (Se ). This implies
that the relationship between the non-equilibrium coefficient and the effective
saturation might be adequate for a certain range of saturation. The discrepancy
between the measurement and the reproduced data may also suggest that the
closure equation may need to be modified by including some other terms. What
should be the correct relationship between τ and the effective water saturation
is not clear yet and more investigation in this regard is needed. Moreover, an
inverse modelling approach and parameter optimization may prove helpful.
BIBLIOGRAPHY
151
Bibliography
Berentsen, C., S. Hassanizadeh, and O. Bezuijen, A.and Oung (2005), Modelling
of two-phase flow in porous media including non-equilibrium capillary pressure
effects, Proceeding Computational Methods in Water Resounces.
Burdine, N. (1953), Relative permeability calculations from pore size distribution data, Transaction of the American Institute of Mining and Metallurgical
Engineers, 198.
Hassanizadeh, S., and W. Gray (1990), Mechanics and thermodynamics of multiphase flow in porous media, Advances Water Resources, 13.
Hassanizadeh, S., and W. Gray (1993a), Thermodinamic basis of capillary pressurein porous media., Water Resources Research., 29.
Hassanizadeh, S., and W. Gray (1993b), Towards an improved description of the
physics of two-phase flow., Advances Water Resources, 16.
Kalaydjian, F. (1992), Dynamic capillary pressure curve for water/oil displacement in porous media, theory vs. experiments, Proc. SPE Conference, Washington DC. SPE, Brookfield, CT.
Chapter
8
Pc − S − awn relationship in a
2D micromodel1
8.1
Introduction
At the microscale, capillary pressure of an interface is given by Young-Laplace
equation:
pc = σnw (K1 + K2 )
(8.1)
where σnw is the interfacial tension between the wetting and nonwetting phase,
and K1 and K2 are the principal curvatures of the surface. This Equation 8.1 describes the capillary pressure at the pore scale and it is valid independently of the
regime in which it is defined; thus under equilibrium and non-equilibrium conditions. Experimental studies at the microscale and macroscale have shown an
hysteretic behavior of the capillary pressure-saturation relationship when reversing the process from drainage to imbibition and viceversa. Thus, the capillary
pressure in a two-phase system or multiphase system is not dependent only on
the saturation. Hassanizadeh and Gray (1990), Hassanizadeh and Gray (1993)
proposed that the capillary pressure-saturation relationship is a two-dimensional
projection of a more extensive functional dependence and a third variable is
needed to explicitly defined the state of the system. They proposed that the
third variable is the specific interfacial area (total interfacial area per unit vol1 Prepared for submission to Water Resources Research, S.Bottero, L.J.Pyrak-Nolte,
S.M.Hassanizadeh
153
154
Chapter 8. Pc − S − awn relationship in a 2D micromodel
ume), awn . The relationship can be written as:
pc = f (awn , Sw )
(8.2)
Reeves and Celia (1996), in a pore network model to study this relationship
based on a personal communication with S.M.Hassanizadeh suggested, that a
better description is provided by the following formula:
awn = f (pc , Sw )
(8.3)
In this section we investigate experimentally the functional relationship between
capillary pressure, pc , wetting phase saturation, Sw , and specific interfacial area
between the wetting and nonwetting phases, awn . The pc − Sw − anw relationship is studied for drainage and imbibition processes under equilibrium and nonequilibrium conditions in a two-dimensional micromodel. Similar experiments
were performed by Chen (2006a), Chen et al. (2007), Pyrak-Nolte et al. (2008)
under static flow condition. However, no experimental studies have been carried
out yet to investigate pc − S − awn functional relationship under non-equilibrium
conditions.
8.2
8.2.1
2D Micromodel Experiments
Experimental set-up
Various experimental methods have been employed to measured specific interfacial area such as synchrotron-based x-ray micro-tomography (Culligan et al.,
2004, 2006), and interfacial tracers (Saripalli et al., 1997; Annable et al., 1998;
Kim et al., 1999a, 1992; Anwar et al., 2000; Rao et al., 2000; Schaefer et al.,
2000; Costanza-Robinson and M.L., 2002; Chen and Kibbey, 2006), photo luminescent volumetric imaging (Montemagno and Gray, 1995). In this study,
we use two-dimensional micro-model to investigate capillary pressure-saturationspecific interfacial area relationship. A general review of 2D micro-model is given
by Giordano and Cheng (2001). A detailed description of the procedure for performing optical lithography is given in the manufacturer’s manual Shipley (1982)
and Thompson et al. (1994). A detailed description on the procedure to generate a micromodel sample is given in Cheng (2002), Cheng et al. (2004), Chen
(2006a) and Chen et al. (2007). The fluids chosen for this series of experiments
were decane, as the wetting phase, and the nitrogen gas as nonwetting phase. In
8.2. 2D Micromodel Experiments
155
Figure 8.1: Schematic representation of the experimental set-up (no to scale).
Table 8.1: Physical properties of decane at 24.5◦ C.
Properties
Surface tension
Viscosity
Contact angle on glass
Contact angle on photoresist
Value
2.474
0.855
4.4
4.1
Unit
mN/m
cp
degrees
degrees
Table 8.1, physical properties of decane liquid at 24.5◦C are listed. A schematic
representation of the experimental set-up is shown in Figure 8.1. It consists of a
nitrogen reservoir connected to a pressure regulator (Mathesontrigas, Model 81
Series, Dual-Stage) and to a cylindrical container of 20 cm diameter and 30 cm
length that enables maintenance of a desired constant nitrogen pressure at the
inlet. Two pressure transducers (Omega PX5500C1-050GV) connected to a multimeter (Ketlyn 2000) recorded the nitrogen pressure while a third transducer
recorded decane pressure at the outlet. The pressure transducers were calibrated
to convert electrical potential (in mV /V ) to pressure (in Pa). Calibration curve
for gas transducers were:
P T1 = 1.11618 · mV /V − 0.141574
(8.4)
156
Chapter 8. Pc − S − awn relationship in a 2D micromodel
P T2 = 1.15487 · mV /V + 0.40615
(8.5)
and for decane transducer was:
P T3 = 1.1917 · mV /V
(8.6)
To reduce the variation of pressure due to temperature fluctuation in the room,
connections and metal tubing were wrapped in a glass fibre. The images were
taken by using a Qimaging Retica EX CCD camera through an Olympus microscope with a 16x objective. The micromodel was placed in a brass holder under
six o-rings, four at the corner of the sample and two at the inlet and outlet to
avoid leak of decane or nitrogen. The micromodel was then covered by a plexiglass layer 3 mm thick. The holder with the micromodel on it was then fixed to
the plate of the microscope.
8.2.2
Fabrication of the micromodel
The micro-model consists basically of two glass cover slides bonded together,
between which is located the micro-model structure. The micro-model has three
regions: the wetting and nonwetting phase reservoirs and the pore-structure,
see Figure 8.2. One glass slide is covered with a thin photoresist layer in which
the pattern and reservoir are fabricated. As shown in Figure 8.2, within each
reservoir four pillar are made to prevent deformation of the two glass plates.
The second glass slide contains two holes of 1 mm diameter that serve as inlet
and outlet. The fabrication of the micromodel is divided into three main steps:
1) preparation of glass slides 2) optical lithography 3) bonding.
Preparation of the glass slides
The glass slides were from Corning Labware and Equipment 18 mm x 18 mm
and 0.25 mm thick. During preparation to reduce the contamination of the
micromodel, the glass slides were carefully cleaned first by immersing them in
a small container of acetone and then shaking them in an ultrasonic bath for
5 minutes. This removed oil and other organic matter. The glass slides were
then immersed into a container of alcohol and shaken in the ultrasonic bath for
another 5 minutes. The glass slides were then blown with nitrogen gas. The
glass slide was coated with photoresist (Shipley 1827 Microposit Photoresist) by
adding 8-9 drops of photoresist and spinning it at 5000 rpm for 30 seconds. The
thickness of the photoresist was controlled by its viscosity and the spin rate. The
desired viscosity of photoresist was obtained through dilution with a thinning
solution (Shipley Microposit Type Thinner).
8.2. 2D Micromodel Experiments
157
Figure 8.2: Schematic representation of the micro-model
Optical Lithography
The micro-model was fabricated using optical lithography. The first step was
to fabricate the inlet and the outlet reservoir. A mask was made of the pillars
and the reservoirs. The mask was placed in direct contact with the photoresist
and exposed to a blue light. This causes a photochemical reaction within the
photoresist making the region exposed to light soluble in a special solution called
developer, (Shipley, 1982). The unexposed photoresist was not soluble, so after
development the photoresist slide contained a negative image of the original light
pattern (mask). By this method, the inlet and outlet patters had the same size
as the mask. A second step was used to construct the desired pore-structure in
the channel between the inlet reservoir and the outlet reservoir. A second mask
was made that contained the desired pore-structure. The transparent mask was
then inserted in the optical path of the microscope (Axioplan Universal Microscope, D-7082, Carl Zeiss) and projected from the mask onto the photoresist
158
Chapter 8. Pc − S − awn relationship in a 2D micromodel
by exposing it to a vis-UV illumination. In this case the projected pattern had
a reduced size compared to the real size of the pattern in the mask. For the
micro-models in this study, we used Shipley photoresist types 1805 and 1827
with their standard developers . The micro-model was fabricated with pore networks covering 600µm x 600µm area. The pore structure for the micromodel
was generated using a standard random continuum percolation model, see Nolte
and Pyrak-Nolte (1991).
8.2.3
Experimental procedure
Preliminary experiments were performed to establish the entry pressure of the
micromodel and the breakthrough pressure, which was used in determining the
number of equilibrium step and the maximum pressure for each (scanning) curve.
The micromodel was initially fully saturated with decane (wetting phase). For
drainage, the nonwetting phase (nitrogen) pressure was increased at the inlet.
Then for imbibition, the nitrogen pressure at the inlet was decreased, allowing the
decane to flow back into the micromodel and expelling the nitrogen. Drainage
and imbibition experiments were performed under both equilibrium and nonequilibrium conditions. With non-equilibrium condition, we refer to the case
that the fluid-fluid interfaces were moving, and both saturation and pressure
were changing with time. We use the term ’equilibrium’ to indicate both static
and steady-state flow conditions. Before drainage experiment started, it was
important to ensure that the nitrogen inlet region was completely filled by nitrogen. It was observed that decane, due to its high wettability, tended to occupy
the corners of the nitrogen reservoir and the roughnesses around the nitrogen
inlet. The presence of decane in the nitrogen inlet area could cause some issues
during drainage processes. For example it was observed that by increasing the
nitrogen pressure a bubble of decane, trapped in the corners of the nitrogen
reservoir, flowed inside the micromodel pattern. This looked like an imbibition
event within a drainage process. Obviously this would have given erroneous interpretation to the data if not taken into account. In other to avoid this, the
nitrogen pressure was kept much lower than the entry pressure of the micromodel
for 40 minutes. By applying this pressure the nitrogen entered slowly only the
inlet reservoir without entering the pattern, such that the decane was forced to
flow out of this region. During equilibrium experiments, the nitrogen pressure
at the inlet was increased or decreased in small steps, typically 200 Pa. At each
pressure increment, the system was allowed to equilibrate before changing the
nitrogen pressure to the subsequent pressure step. Typically, equilibrium was
reached in 5-7 minutes. The system was considered in equilibrium when no variations in the curvature of the wetting-non-wetting interface and of the pressures
8.3. Data image analysis
159
were observed. During drainage, the nitrogen was allowed to flow into the micromodel in steps. At some pressure, it reached the outlet. After breakthrough,
with the nitrogen pressure kept constant, steady-state was reached. Thus, two
flow states could be distinguished: static (before breakthrough) and steady state
(after breakthrough). At each equilibrium step, the inlet nitrogen pressure, pressure of decane at the outlet reservoir, and the room temperature were recorded.
At the same time, an image of the micro-model pattern was taken. Various
equilibrium scanning and drainage imbibition curves were generated by reversing the pressure step at different equilibrium states. A scanning drainage was
incrementally continued until reached the main imbibition curve and a scanning
imbibition was incrementally continued until it reached main drainage curve.
For the non-equilibrium drainage experiments, contrary to equilibrium ones,
the nitrogen pressure was increased in one large step. The inlet pressure was kept
constant throughout the entire experiment. First, the valve at the inlet of the
micro-model was closed (see Figure 8.2). Then the pressure of the nitrogen tank
was increased to a desired value. The nitrogen pressure was allowed to reach a
constant value everywhere in the system, e.g. in the constant pressure container
and in the metal tubes. Once a constant pressure was obtained, the inlet valve
was opened (see Figure 8.1). The micromodel was initially fully saturated with
decane. The nitrogen flowed into the micromodel displacing the wetting phase.
The nitrogen was allowed to reach the opposite side of the micromodel. During
imbibition, the pressure from the pressure tank was drastically decreased opening
a second valve (see Figure 8.1). The external pressures and the digital images
were acquired every 0.5 second. The camera was able to acquire images faster
that 0.5 second. But at the same time also pressures had to be recorded and
this increased the acquisition time.
8.3
Data image analysis
Each image was analyzed to extract the phase saturation, interfacial area per
volume, awn , and curvature of the interfaces. The image analysis program was
written in IDL. Details about the program are given in Pyrak-Nolte et al. (2008).
Here, for the sake of completeness a summary description will be given.
At the beginning of each experiment, the micromodel was filled with nitrogen
and a image of the pore structure was obtained. From this image, a binary image
was extracted (shown in Figure 8.3 a), which served as the base pattern image in
the analysis of subsequent images in that experiment. Then for each digital image
taken during drainage or imbibition three different binary images were extracted
(see the example shown in Figure 8.3): one showing the region of photoresist,
160
Chapter 8. Pc − S − awn relationship in a 2D micromodel
Figure 8.3 a, one with nitrogen Figure 8.3 b, and the other with only decane,
Figure 8.3 c. A combined image of the three binary images is shown in Figure
8.3 d. After all three phases were distinguished, the average saturation of a given
phase over the micromodel was calculated counting the pixels for that fluid phase
and dividing by the total number of pore space pixels. Because the images are
Figure 8.3: a) binary photoresist phase; b) binary nitrogen phase; c) binary
decane phase; d) composite image
two dimensional, an interfacial length per unit area was calculated instead of an
interfacial area per volume. This is due to the difficulty in imaging the hidden
curvature of the interface across the depth of the micromodel. In any case, we
refer to the interfacial length per unit area as specific interfacial area. It must
be noted that in a pore where nitrogen was advancing, two types of interface
were found. A main interface spanning over the width of a pore; we refer to
this as main terminal interface; and interfaces between nitrogen and the decane
film left in the corners of the pores. These are referred to as film interfaces. In
8.3. Data image analysis
161
fact these films interface are present in the whole nitrogen filled region. In the
calculation of the specific interfacial area, only the main terminal interfaces are
considered. A Sobel edge detector was used to identify interfaces between the
phases. The wetting-nonwetting interface length, Lwn , was calculated from the
following equation:
Lwn = (Lw + Ln − Ls )/2
(8.7)
where: Lw , Ln and Ls are the total lengths of boundary of the wetting (decane),
nonwetting (nitrogen), and solid phases, respectively. The specific interfacial
area was then found by dividing Lwn by the total area (A) of the micromodel:
awn =
Lwn
A
(8.8)
Chen (2006a) pointed out that the error in saturation and interfacial area from
this analysis method depended on the resolution of the image and also on the
curvature of fluid-fluid interface. When high resolution images were used, the
error in saturation was less than 1%. The error estimated in interfacial area was
at most 10 − 15%. Other methods exist for determining interfacial area Dalla
et al. (2002); McClure et al. (2007), that might have smaller errors. The average capillary pressure was calculated from an equation similar to 8.1 in which
an average curvature was used. Because we neglected the hidden curvature of
interfaces (as it was not measured) we replaced 8.1 with
p̄c = σnw K̄
(8.9)
Now, the interfaces in a given single image did not have all the same curvature; not even under static conditions. This was partly due to the limitations
in imaging. In fact for any single image, we had a distribution of curvatures.
So a representative curvature for the whole micromodel had to be calculated
and inserted into the Equation 8.9. The representative curvature K̄ for any
single image could be calculated in two ways. Obviously the capillary pressure
will be different depending on the method chosen. One way was to determine
the average curvature Kaver of the distribution of curvatures. The other way
was to use the mode value of curvature Kmode(i.e. the most frequent value of
the distribution). Thus, we distinguished two values for the average capillary
pressure:
Pcaver = σnw Kaver
(8.10)
162
Chapter 8. Pc − S − awn relationship in a 2D micromodel
Pcmode = σnw Kmode
(8.11)
Pyrak-Nolte et al. (2008) showed that, under static conditions, the two ways
of calculating capillary pressure gave similar results. However, the two values
of capillary pressures are not necessarily always equal. The average curvature
could be different from the most frequent curvature for an image.
8.4
Pc − sw − awn surfaces
Hassanizadeh and Gray (1993) state that the hysteresis in capillary pressuresaturation relationship is probably observed because we work with the projection
of a Pc − S − awn surface on the capillary pressure-saturation plain. Hence,
according to this theory, it is probable that Pc − S − anw relationship would
not be hysteretic, and all points from all processes (drainage or imbibition) and
under all conditions (equilibrium and non equilibrium) would lie on the same
surface. Chen et al. (2007) performed (primary as well as scanning) drainage
and imbibition experiments in a 2D micromodel under static conditions. They
found that the two surfaces, one for all drainage points and the other for all
imbibition points, were the same to within 10%. Whether that holds also for
processes under non-equilibrium conditions is not studied yet. The reader should
remember that in our experiments what is actually determined is the average
capillary pressure based on the local Pc at the interface of the nonwetting and
wetting phases and not the difference in average pressure of the two fluids. So
the question arise whether Pc − S − awn surface measured under equilibrium
and non-equilibrium conditions would be the same. To establish whether the
Pc − S − awn data points from drainage and imbibition experiments under both
equilibrium and non-equilibrium lie on the same surface, we have generated four
pc − S − awn surfaces, one for each process and condition. These surfaces are
then compared one by one.
A surface was generated with Matlab code by using two functions: 1) MESHGRID and 2) GRIDDATA. The data point S − pc did not fall on a regular grid.
MESHGRID created a regular grid in a x,y plain from a irregularly distributed
points, in this case S, Pc . The limits x and y of this regular grid were specified by
the maximum and minimum values of S and Pc , respectively. The limits of the
capillary pressure and saturation were chosen to be the same for both process
drainage and imbibition processes. The output of the MESHGRID was then
used in GRIDDATA along with the data points Pc , S, awn . The GRIDDATA
8.5. Results and discussions
163
procedure returned a regular grid of interpolated z value, in this case anw . The
’v4 method’ was used to interpolate the data points. The ’v4 method’ is based
on Radial Basis function and on Green’s function (see Matlab). It is important
to pointed out that GRIDDATA function does not scale the axes. Thus, in our
case because of the large magnitude of the specific interfacial area and capillary pressure relative to the limits of the saturation (0 − 1) we would see only
variation on the specific interfacial area and not in the plain Pc − S. Therefore,
saturation, capillary pressure, and specific interfacial area are normalized with
the maximum value of each variable.
8.5
8.5.1
Results and discussions
Equilibrium experiments
In Table 8.2, all drainage and imbibition experiments under equilibrium condition along with the maximum external pressure reached at the end of each
drainage precesses and the number of images analyzed are listed. As was previously explained, the experiments were performed by increasing the external
pressure in steps of 200 Pa and waiting between step for the system to equilibrate. The nonwetting phase in this setup was allowed to reach the opposite
end site of the micromodel, i.e., breakthrough. Thus, two flow conditions could
be distinguished: static (before breakthrough) and steady state (after breakthrough). Here the term ’equilibrium’ refers to the state of the interfacial area
and the phase saturation, independent of the flow regime. Figure 8.4 shows the
equilibrium external pressure-saturation curves. Given the fact that the wetting phase reservoir pressure was zero, the external pressure can be seen as the
pressure difference between the nitrogen gas and the decane phase reservoirs.
It was observed that after each drainage-imbibition cycle there is no a residual
nonwetting phase saturation. Thus, after imbibition, the pore voids are again
filled only with decane. This is caused by the geometry the micromodel. In fact
there is not a clear distinction between pore throats and pore bodies. Another
factor which might contribute to no residual saturation is the high wettability of
the photoresist to decane. In Figure 8.4, at high saturation, the drainage curves
in test4 lie lower than the drainage curves obtained in the series test5 and test6.
For test4 experiments, the inlet and outlet were switched around. This was done
by turning the micromodel by 180 degrees on the horizontal plane. Although the
pattern obviously remained unchanged, a lower external pressure was needed in
test4 in order to reach the same saturation as in test5 and test6. For this reason
the test4 was not considered in further analysis.
164
Chapter 8. Pc − S − awn relationship in a 2D micromodel
45
test4a loop1 D
test4a loop1 I
test4a loop2 D
test4a loop2 I
test4a loop3 D
test4a loop3 D
test4b loop1 D
test4b loop1 I
test4b loop2 D
test4b loop2 I
test5b D
test5b I
test5c D
test5c I
test5d D
test5d I
test6a D
test6a I
test6b D
test6b I
test6c D
test6c I
test6d D
test6d I
External pressure, [kPa]
40
35
30
25
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Decane Satuation, [−]
0.8
0.9
1
Figure 8.4: Primary and scanning drainage and imbibition points under equilibrium condition (full hysteresis loop).
Figure 8.5 a shows a comparison between external pressure difference (symbols with a continuous line) and the average capillary pressure calculated through
average curvature (symbol without line) during drainage. Under static conditions the external pressure difference and the capillary pressure are the same.
But after breakthrough (between saturation 0.7 and 0.6), an internal capillary
pressure drop was observed. This is because after breakthrough a relaxation of
the curvatures of the fluid-fluid interfaces occurred which resulted in a decrease
of the local capillary pressure and thus of the average capillary pressure. Figure
8.5 b shows the specific interfacial area versus decane saturation for all primary
and scanning drainage processes. The interfacial area first increased with the
decrease of decane saturation to a maximum peak and then it decreased again
when the saturation approached the irreducible saturation. The reader should
remember that in the calculation of the specific interfacial area only the terminal interfaces were considered and interfaces related to the films of the wetting
8.5. Results and discussions
pressure, [kPa]
45
165
a
40
35
30
25
0.2
0.4
0.6
Decane saturation, [−]
0.8
1
Specific interfacial area, [1/m]
4
2
x 10
test5b D
test5c D
test5d D
test6a D
test6b D
test6c D
test6d D
b
1.5
1
0.5
0
0.2
0.4
0.6
Decane saturation, [−]
0.8
1
Specific interfacial area, [1/m]
4
2
x 10
c
1.5
1
0.5
0
0
1000
2000
3000 4000
pc−aver, [Pa]
5000
6000
7000
Figure 8.5: a) Comparison between external pressure difference (symbols with
a continuous line) and average capillary pressure (only symbols) versus decane
saturation; b) Specific interfacial area versus decane saturation; c) Specific interfacial area versus average capillary pressure. The plots are from equilibrium
drainage experiments.
166
Chapter 8. Pc − S − awn relationship in a 2D micromodel
phase on the photo resistance were not included. This behavior is in agreement
with the numerical models suggested by Givirtzman and Roberts (1991), Reeves
and Celia (1996); Held and Celia (2001); Joekar-Niasar et al. (2008). Their
results suggested a parabolic anw − S relationship when the contribution to the
interfacial area was only due to terminal fluid-fluid interfaces. When films were
ignored, wetting phase saturation was observed to be zero at the end of drainage,
causing a zero interfacial area. A monotonic behavior has been observed when
both fluid-fluid interfaces and films were considered. This was reported in experiments by Chen (2006b) and Chen and Kibbey (2006). They performed a
series of experiments at the column scale under slow-dynamic (pseudo-static)
conditions to investigate pc − S − anw relationships. This monotonic behavior was also observed numerically by Cary (1994), Silverstein and Fort (1997,
2000), Or and Tuller (1999), Tuller et al. (1999) , Joekar-Niasar et al. (2008).
When both terminal interfaces and wetting films were included, the relationship
between interfacial area and saturation was characterized by a monotonic increase of interfacial area with decreasing wetting phase saturation. Figure 8.5 c
shows the specific interfacial area versus average capillary pressure. In all cases
the specific interfacial area increases with the decrease in the decane saturation. Figure 8.6 a shows the pressure difference (points with a continuous line)
and the average capillary pressure (only symbols) during imbibition process. At
low decane saturation, the capillary pressure is much lower than the external
pressure difference. The capillary pressure at low saturation corresponds to the
last equilibrium point on the drainage process. The variation of the capillary
pressure from the initial condition to the end of the imbibition is quite small.
At the end of imbibition, the decane saturation return to unity. Thus, there
was no trapping of the nonwetting phase in the micromodel. Figure 8.6 b shows
the specific interfacial area versus decane saturation for imbibition. The specific
interfacial area shows a non-monotonic behavior but with the peak occurring
at a decane saturation different from drainage peak (compare to Figure 8.5 b).
Moreover, the higher specific interfacial area for imbibition is grater than for
drainage. Figure 8.6 c shows the specific interfacial area versus average capillary pressure. For all cases, a monotonic increase in specific interfacial with
decreasing capillary pressure is observed.
Figures 8.7 a and b show Pc − S − anw equilibrium data points along with
the interpolated surface for drainage and imbibition processes, respectively. Note
that the axes are normalized to the maximum value of each variable. The same
maximum values were used for both drainage and imbibition. Ideally, these two
surfaces should coincide. Joekar-Niasar et al. (2008) fitted the Pc − S − anw
data points obtained from a pore-network model by a second-order polynomial
8.5. Results and discussions
167
45
pressure, [kPa]
a
40
35
30
25
0.2
0.4
0.6
Decane saturation, [−]
0.8
1
Specific interfacial area, [1/m]
4
2
x 10
b
test5b I
test5c I
test5d I
test6a I
test6b I
test6c I
test6d I
1.5
1
0.5
0
0.2
0.4
0.6
Decane saturation, [−]
0.8
1
Specific interfacial area, [1/m]
4
2
x 10
c
1.5
1
0.5
0
0
1000
2000
3000
4000
pc−aver, [Pa]
5000
6000
Figure 8.6: a) Comparison between external pressure (symbols with a continuous line) and average capillary pressure (only symbols) versus decane saturation;
b) Specific interfacial area versus decane saturation; c) Specific interfacial area
versus capillary pressure. The plots are from equilibrium imbibition process experiments.
168
Chapter 8. Pc − S − awn relationship in a 2D micromodel
IVA/ max IVA, [−]
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
Pc/ max Pc, [−]
1
0.8
0.6
0.4
0.2
Sw/ max Sw, [−]
IVA/ max IVA, [−]
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
Pc/ max Pc, [−]
1
0.8
0.6
0.4
0.2
Sw/ max Sw, [−]
Figure 8.7: a) Pc −S−awn surface fitted to normalized data points for equilibrium
drainage experiments; b) Pc − S − awn surface fitted to normalized data points
for equilibrium imbibition experiments.
8.5. Results and discussions
169
Table 8.2: Drainage and imbibition experiments under equilibrium condition.
N
1
2
3
4
5
6
7
8
9
10
11
12
Test
4a loop1
4a loop2
4a loop3
4b loop1
4b loop2
5b
5c
5d
6a
6b
6c
6d
Process
drain/imbib
drain/imbib
drain/imbib
drain/imbib
drain/imbib
drain/imbib
drain/imbib
drain/imbib
drain/imbib
drain/imbib
drain/imbib
drain/imbib
n. of analyzed images
22
29
33
48
62
98
71
128
35
35
50
24
Max pressure (kPa)
33.92
34.00
34.32
34.53
34.36
36.94
38.69
42.89
36.11
38.14
41.48
35.31
surface. Their results were specific to a simple and well structured porous media. This does not apply to our case as the micro-model’s pattern was highly
non-structured. To compare the two surfaces values of specific interfacial area
obtained from the two surfaces, at a given Pc − S points were divided to each
other. The closer the ratio to unity is, the closer the two surfaces are to each
other. Because data points of imbibition and drainage were not available for the
same Pc − S points, values of interfacial area for a given Pc − S point were found
by interpolation. Thus, we could take a measured Pc − S point from drainage
data and find the corresponding value of interfacial area for imbibition surface by
interpolation. Alternatively, we could choose measured Pc − S points from the
imbibition data and find the corresponding interfacial area for drainage by interpolation. A similar approach was used by Chen et al. (2007). The ratio of the
measured drainage interfacial area to the interpolated imbibition interfacial area
varied and a histogram of ratio is show in Figure 8.8 (designated by Histogram I
in light gray on the figure). In the same figure, the histogram of the ratio between
measured imbibition interfacial area to the interpolated drainage interfacial area
is given as Histogram II (shown in dark gray). If the ratio is one, the measured
and interpolated interfacial areas are the same. The mode for Histogram I is
at 0.9 and the average ratio is 0.88 (see also Table 8.3). For histogram II, the
maximum mode is at 1 and the average ratio is 1.28. The values are in within the
measurement error of 10 − 15%. Thus the Pc − S − awn surface under drainage
and imbibition may be considered to be the same within the measurement error. These results support the conjecture put forward by Hassanizadeh and Gray
(1993), who suggested that the hysteretic in Pc −S relationship may be modelled
170
Chapter 8. Pc − S − awn relationship in a 2D micromodel
60
IVA−data, Drain / IVA−inter, Imb
IVA−data, Imb / IVA−inter, Drain
50
Counts
40
30
I
20
10
0
0
II
0.5
1
1.5
2
2.5
IVA measured/interpolated
3
3.5
Figure 8.8: Histogram I represents the ratio between measured specific interfacial
awn throughout drainage to the interpolated values on the imbibition surface.
Histogram II represents the ratio between the measured specific interfacial area
under imbibition to the interpolated values on the drainage surface. Both are
related to a equilibrium experiments.
through the inclusion of specific interfacial area as in this relationship. Similar
results were reported by Cheng et al. (2004); Chen et al. (2007) for experiments
carried out in a similar 2D-micromodel. They found that Pc − S − awn relationship under imbibition was non-hysteretic to within 5%. Chen et al. (2007)
compared both drainage and imbibition Pc − S − awn surfaces and found that
they are non-hysteretic in within 10%, which is the same magnitude of the measurement error from image analysis. Reeves and Celia (1996) investigated the
role of the interfacial area by a pore-network model. They found similarity in the
shape of values drainage and imbibition surfaces but there was significant difference with regards the maximum values of interfacial area. They suggested that
different results could have been obtained if different mechanisms were adopted.
For example, the sequence in which the scanning curves were generated could
be an explanation for the difference between the two surfaces. Held and Celia
(2001) modified the pore-network model of Reeves and Celia (1996). They employed a different imbibition mechanism and included the effect of snap-off of the
nonwetting phase. Their results suggested that the Pc − S − awn surfaces generated for drainage and imbibition scanning were the same to within 1.5% when
the optimized snap-off and local configuration parameters were selected such
8.5. Results and discussions
171
Table 8.3: Mode and average ratio between measured and interpolated specific
interfacial area.
Histogram
I
II
III
IV
V
VI
VII
VIII
ratiof data/interpolated
data, eqD /int, eqI
data, eqI /int, eqD
data, n.eqD /int, n.eqI
data, n.eqI /int, n.eq.D
data, n.eqD /int, eqD
data, n.eqI /int, eqI
data, n.eqD /int, eqI
data, n.eqI /int, eqD
mode
0.9
1
1
0.9-1.20
1.2
1
0.9
2.2
average
0.88
1.28
1.08
0.92
1.30
1.25
1.31
1.82
that to minimize the hysteresis. Joekar-Niasar et al. (2008) also investigated
the relationship Pc − Sw − awn throughout drainage and imbibition process in a
pore-network model. They found that difference between drainage and imbibition surfaces decreased for pore-structures in which the distinction between pore
throats and pore voids size decreased. In column experiments, Chen (2006b)
however, found a non-unique Pc − S − awn surfaces for the system they studied
particularly when primary drainage was included. The difference between our
results and their results could be due to the fact that they accounted for the
interfacial areas of wetting films, which are difficult to track in micromodel, or
due to differences between the drainage behaviors of 2D and 3D porous media.
8.5.2
Non-equilibrium drainage and imbibition experiments
Table 8.4: Non-equilibrium experiments.
Test
test7b
test7b
test7d
test7d
test7e
test7e
test7l
test7l
Process
drainage
imbibition
drainage
imbibition
drainage
imbibition
drainage
imbibition
analyzed images
779
377
728
603
938
886
1092
33
Max pressure (kPa)
34.00
34.00
34.53
34.53
38.14
38.14
38.69
38.69
Table 8.4 lists the non-equilibrium drainage and imbibition tests, with the
number of images analyzed and the corresponding external pressures applied
172
Chapter 8. Pc − S − awn relationship in a 2D micromodel
at the inlet. Figures 8.9 a and 8.9 b show the Pc − S − awn surfaces fitted to
drainage and imbibition data, respectively. Throughout the drainage process,
the decane saturation is higher than the decane saturation under equilibrium
condition (see Figures 8.9 a and b and Figures 8.7 a and b). This occurs because during non-equilibrium conditions the external pressure is higher than the
local entry pressure of a pore throat. Thus, the nitrogen flowed through the
micromodel in a channel towards the opposite side displacing less wetting phase
than in the equilibrium case. In the equilibrium experiments, the pressure was
increased in small steps thus the micromodel was slowly filled, and a larger pore
voids area could be occupied by the nonwetting phase. For the non-equilibrium
case, we investigated whether the drainage and imbibition data points lied on
the same pc − S − awn surface. The same approach as for the equilibrium
case was adopted. Figure 8.10 shows the histogram of the ratio of measured
drainage interfacial area to the interpolated imbibition value and is designated
as Histogram III. The mode value is 1.0 which indicates that at majority of
points from the drainage and imbibition surface are coincident. This suggests
that there is only one Pc − S − awn surface representative for drainage and
imbibition for this specific fluids and this specific micro-model structure under
non-equilibrium condition. The average ratio is at 1.08. In the same Figure 8.10,
histogram IV represents the frequency of the ratio between the imbibition data
and the interpolated data over the drainage surface. In this case, two modes can
be distinguished: one at 0.9 and 1.20. The average ratio is 0.92. Thus also in the
case of non-equilibrium condition the two surfaces, drainage and imbibition, can
be considered coincident within the measurement errors. The question remains
whether the equilibrium and non-equilibrium data points fall on the same surface
such that we can state that only one Pc − S − awn surface exist independently of
the processes and of the conditions and dependent only on the combination of
fluid-fluid and pore-structure. Therefore, the ratio of interfacial area measured
under non-equilibrium drainage to the interfacial area obtained for surface fitted
to the equilibrium drainage points was calculated. The same was done for the
imbibition surfaces. In Figure 8.11 histogram V and histogram VI represent the
frequency of the ratios for drainage and imbibition, respectively. It can be seen
that for imbibition the mode is at 1.05 and the average is 1.30. In histogram
V the mode is at 1.25 and the average is 1.25. These ratios are of the same
order observed for the equilibrium and the non-equilibrium points were considered separately. Thus, within the error measurements the non-equilibrium and
equilibrium points all lie on the same surface Pc − S − awn . Finally points from
non-equilibrium drainage are compared to points from equilibrium imbibition,
as well as non-equilibrium imbibition compared to equilibrium-drainage. The
8.5. Results and discussions
173
“
Figure 8.9: a) Pc − S − awn surface fitted to normalized data points for nonequilibrium drainage experiments; b) Pc − S − awn surface fitted to normalized
data points for non-equilibrium imbibition experiments.
174
Chapter 8. Pc − S − awn relationship in a 2D micromodel
1200
data D / inter I; dynamic
data I / inter D; dynamic
1000
Counts
800
600
III
400
200
IV
0
0
0.5
1
1.5
IVA measured/interpolated
2
2.5
Figure 8.10: Histogram III represents the ratio between measured specific interfacial awn under drainage to the interpolated values on the imbibition surface.
Histogram IV represents the ratio between the measured specific interfacial area
under imbibition to the interpolated values on the drainage surface.
corresponding histograms are shown in Figure 8.12. In the histogram VII the
mode is 0.9 and the average is 1.31 which is comparable with the previous results. In histogram VIII the mode is at 2.2 and the average is 1.82. This last
case excludes the existence of a unique Pc − S − awn surface for equilibrium and
non-equilibrium conditions. Thus, further studies are needed in this regard.
8.6
Summary and Conclusions
A series of drainage and imbibition experiments in a 2D micro-model were carried
out to investigate the role of specific interfacial area in modelling capillarity in
two phase flow. Under equilibrium conditions at each pressure step external
pressure and images of fluid distributions were recorded. The fluids were allowed
to flow into the micro-model and reach its opposite side. Thus, two different
flow conditions could be distinguished: static and steady-state. With the term
equilibrium we refer to the state where fluid-fluid do not move and saturation
remain constant. In non-equilibrium experiments, pressures and images were
8.6. Summary and Conclusions
175
1000
IVA−data, drain dynamic / IVA−inter, drain static
IVA−data, Imb dynamic / IVA−inter, Imbib static
900
800
700
Counts
600
500
V
400
300
200
VI
100
0
0
0.5
1
1.5
2
IVA measured/interpolated
2.5
3
Figure 8.11: Histogram V represents the ratio between measured specific interfacial awn during nonequilibrium drainage to the interpolated values on the equilibrium drainage surface. Histogram VI represents the ratio between the measured
specific interfacial area under nonequilibrium imbibition to the interpolated values on the equilibrium imbibition surface. Both are related to a nonequilibrium
experiments.
taken every 0.5 second without waiting for the interface to equilibrate. The
images were analyzed by using IDL software to extract the average saturation,
fluid-fluid curvature, and the specific interfacial area. In these calculations only
the interface spanning over the width of a pore was considered. We refer to
this as a main terminal interface. Whereas the interface between wetting and
nonwetting phase film left in the corners of the pores are not considered.
To establish whether the data points all fall in the same Pc − S − awn surface,
a ratio approach was used. The measured specific interfacial area were divided
by the specific interfacial area obtained from the interpolated surfaces. The
closer to unity the ratio is, the closer are the two surfaces. The results from the
equilibrium drainage and imbibition surfaces show an average ratios of 0.8 and
1.28 which are in the range of the measurement error. In fact, Chen (2006a);
Chen et al. (2007) estimated an error of 10 − 15% in the calculation of the
interfacial area based on the method they used to extract these values from the
images. Thus, within the measurement error, drainage and imbibition surfaces
176
Chapter 8. Pc − S − awn relationship in a 2D micromodel
1200
measured drain. non−equilibrium / interpolated imb. equilibrium
measured imb.non−equilibrium / interpolated drain.equilibrium
1000
Counts
800
600
400
200
VII
VIII
0
0
0.5
1
1.5
2
IVA measured/interpolated
2.5
3
Figure 8.12: Histogram VII represents the ratio between measured specific interfacial awn during nonequilibrium drainage to the interpolated values on the
equilibrium imbibition surface. Histogram VIII represents the ratio between the
measured specific interfacial area under nonequilibrium imbibition to the interpolated values on the equilibrium drainage surface.
can be considered to be the same surface. The results from non-equilibrium
experiments show that the ratio between measured and interpolated value of
specific interfacial area was 1.08 and 0.92. This also indicates that under nonequilibrium conditions independent of the process (drainage or imbibition) the
data points fall on the same surface Pc − S − awn . This is in agreement with the
theory suggested by Hassanizadeh and Gray (1993). The same ratio approach
was used to investigate whether the non-equilibrium and equilibrium data points
under drainage or imbibition conditions fall in the same surface. The average
ratio between measured specific interfacial awn during nonequilibrium drainage
to the interpolated values on the equilibrium drainage surface results 1.31. While
the ratio between the measured imbibition specific interfacial area under nonequilibrium to the interpolated ones under equilibrium imbibition process give
1.25. These ratios are closed to each other and within the measurement error
indicate that the surfaces for drainage and imbibition under equilibrium and
non-equilibrium are the same.
BIBLIOGRAPHY
177
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Chapter
9
Summary and Conclusions
The interest in understanding and properly describing multiphase flow in porous
media has increased immensely over the last decades due to the importance
of many engineering applications such as remediation of polluted sites, paper
production, fuel cells. In this thesis various aspects related to the theory of
two-phase flow in porous media were investigated.
Chapter 2 gives an overview of the fundamentals related to the physical
processes which underlie the flow of immiscible fluids at the micro-scale and
macro-scale.
Chapter 3 focus on the definition of average pressure. Although at first one
may think that defining an average fluid pressure is an easy task, the answer
to a simple question such as: what is the correct averaging pressure, is not trivial. Several averaging operators were introduced and used to obtain average or
upscaled fluid pressures. The most commonly used averaging operator is the intrinsic phase-volume average, which weighs phase pressure with phase saturation
values. An alternative is the simple phase average which average pressure only
over the regions where the phase is actually present. Yet another alternative is
the simple average, assuming that the pressure of phase α is defined everywhere
in the domain of interest and performs the averaging over the whole domain.
The centroid-corrected phase average pressure, or in short the centroid-corrected
pressure was introduced by Nordbotten et al.(2008) as a part of a family of
macroscale pressures. This averaging operator corrects the intrinsic phase average pressure for the distance between the centroid of the averaging volume and
the centroid of the phase.
We have studied the differences among these averaging operators by apply181
182
Chapter 9. Summary and Conclusions
ing them to static equilibrium situations in a vertical homogeneous domain. We
consider both primary and main drainage cases, where the nonwetting phase was
injected through the bottom boundary. At static equilibrium, pressure and saturation distributions were derived for a given nonwetting phase bottom boundary
overpressure ∆P . These distributions were used in the calculation of average
pressure and saturation. An important feature of static equilibrium is that the
total phase potential (i.e. pressure plus gravity potential) is constant over the
whole domain. Therefore, its average will be equal to the same constant. Thus,
we imposed the criterion that the average phase pressure (which is assigned to
the centroid of the averaging volume), plus the gravity potential at the domain
centroid must be equal to the constant phase potential for any given condition.
We have found that the intrinsic phase-volume average pressure, which is commonly employed in averaging studies, results in a gradient in the total phase
potential, i.e. the above criterion is violated. In fact, only the centroid-corrected
operator satisfies this criterion. However, at high saturations, use of the centroidcorrected average can give rise to negative values of the difference between the
average nonwetting and wetting phase pressures. For main drainage, differences
among various averaging operators are significantly less because both phases are
present initially, such that the difference between the centroids of phases and
the middle of the domain are relatively small. After defining average pressures
the question arises: what is actually the average capillary pressure? What is the
proper way of determining capillary pressure at the macroscale? In this study,
in order to investigate the behavior of the capillary pressure-saturation relationship under equilibrium condition, a series of drainage and imbibition experiments
were carried out.
In Chapter 4 the experimental set-up including devices for measuring local
pressures and saturations as well as average saturation over the whole sample,
calibration curves, sample preparation, and the experimental procedure are presented.
In Chapter 5 a procedure to determine an upscaling capillary pressure is
proposed. Common practice to determining capillary pressure is subtracting the
wetting phase pressure from the nonwetting pressure measured at the outside
fluid reservoirs. This is then assumed to be the representative average capillary
pressure for the system fluids-porous medium. However, the external fluid pressure differences does not correspond to the capillary pressure-saturation curve
representative of the fluids-porous media system. In fact, this method yields
an inaccuracy due to gravity effects. For example for a large column gravity
plays a role. So the outcome of the common procedure to measure capillary
pressure-saturation curve is not a real capillary pressure and cannot character-
183
ize fluids-porous medium system. Here instead average capillary pressure was
determined based on average fluid pressures employing various operators mentioned above.
Primary drainage, main drainage and main imbibition experiments were performed in a relatively homogeneous sand column under equilibrium conditions.
The fluids employed were: PCE as nonwetting phase, and de-mineralized deaired water as wetting phase. In a drainage process the column was initially fully
water saturated and PCE was injected from below. The injection pressure was
increased in small increments and kept constant for 24 hours until equilibrium
was reached before changing to the subsequent pressure step. During imbibition
the flow was reversed. The pressure of the injection fluid was decreased in small
steps and water was allowed to flow inside the column from above. Phase pressures and saturation inside the column at various elevations, as well as external
pressure and average saturation were recorded at each equilibrium step.
Various averaging operators introduced above were used to determine the
average of each fluid pressure based on locally measured pressures at four observations points. The average capillary pressure representative for the entire
averaging domain was obtained by subtracting the average wetting pressure
from the average nonwetting pressure. The comparison between averaged capillary pressure-saturation curves shows that the curve based on the centroidcorrected average better approached the reference curve or potential-based Pc −S
curve. However, while the potential-based is valid only under static condition the
centroid-corrected average can be used for flow situation. Moreover, it was noticed that within the measurement error, the local primary drainage capillary
pressure curve is close to the average capillary pressure-saturation curve based
on the centroid-corrected average operator. Thus, under equilibrium condition
and for a homogeneous porous media, the local capillary pressure and the average capillary pressure-saturation curve can be describe by the same functional
relationship Pc (Sw ). During main drainage and main imbibition, the difference
between various average phase pressures based on different averaging operators
was found to be negligible. This is because both phases were present from the
beginning and the centroids of phases and the centroid of the averaging domain
were close to each other.
In Chapter 6 non-equilibrium effects in pressure phase difference are investigated. Capillary pressure saturation relationship is property of the fluidsmedium system and is determined under equilibrium condition. However, this
is commonly used to describe processes that occur under non-equilibrium conditions. Kalaydjian (1992) and Hassanizadeh and Gray (1990) based on thermodynamic approach, suggested that under non-equilibrium conditions the fluids
184
Chapter 9. Summary and Conclusions
pressure difference should be equal to the capillary pressure plus an extra term
proportional to the rate at which the saturation changes. The corresponding
coefficient, τ was said to be a function of saturation. This relationship has been
a source of debate. Several questions arise: how do pressure differences and capillary pressure compare with each other under non-equilibrium condition? What
is the functional relationship τ (Sw )? Does this coefficient scale with the length
of the domain?
A series of dynamic primary drainage experiments were carried out in order to investigate non-equilibrium effects in fluid pressure difference-saturation
relationship. In this case the pressure was increased to a large value at once
and kept constant throughout the experiments. The injection pressures were
20 kPa, 30 kPa, 35, kPa, and 38 kPa. The resulting non-equilibrium pressure
difference-saturation curves is found to lie higher than the capillary pressuresaturation curve. Moreover, the non-equilibrium pressure difference shows a
nonmonotonic behavior with an overshoot which is more pronounced at higher
injection pressure. Average fluids pressure difference and saturation were calculated for a column domain based on local measured pressures and saturation.
As it was already shown for the steady-state case, the intrinsic average operator
is found to overestimate the nonwetting phase average. This averaging effect is
quite pronounced in the non-equilibrium case as the pressures applied are much
larger. The average non-equilibrium pressure difference determined by the centroid corrected average operator still lie above the average capillary pressure.
Moreover for higher injection pressures it also shows a non-monotonic behavior.
The non-equilibrium coefficient τ was calculated at the local scale by three different methods. Although each method showed different trend on the variation
with saturation, the τ value at this local scale is of the order of 105 P a.s. The
non-equilibrium coefficient was also estimated at the column scale by employing average saturation and average phase pressures. The results show that this
coefficient is one order of magnitude higher than the local damping coefficient,
around 106 P a.s. Obviously, it is even larger when the intrinsic phase average
is adopted. However, it should be pointed out that it does not scale with the
length square of the average domain as suggested by Dahle et. al (2005) for a
bundle of tubes.
In Chapter 7 non-equilibrium primary drainage laboratory experiments
were simulated first by employing the traditional two-phase flow equations and
then including the non-equilibrium term as suggested by Kalaydjian (1992) and
Hassanizadeh and Gray (1990). Various functional dependencies of the nonequilibrium coefficient τ on wetting phase saturation, namely, linear, quadratic,
error function and exponential were considered. In order to compare simulated
185
results with experimental data, the nonwetting and wetting phase pressures, as
well as wetting saturation, were averaged over an area comparable with measurement window of the pressure transducers and TDR sensors.
Results have shown that, under non-equilibrium condition, the averaging operator chosen to average phase pressure plays an important role even at sensor
scale. In fact depending on the boundary pressure, large differences in the average pressures can be obtained whenever the intrinsic-phase averaging operator
or the centroid-corrected operator is employed.
It is found that for the injection pressure of 20 kPa, non-equilibrium effect on
fluid pressure differences are not significant. For small value of τ the simulation
results do not differ from the ones performed without the non-equilibrium term.
The experimental data at high injection pressures of 30 kPa, 35 kPa and 38
kPa exhibited a large overshoot of the non-equilibrium pressure difference curve
versus time. Numerical results revealed that this non-monotonic behavior can be
reproduced only if the dynamic term is included in the equations describing twophase flow. Although the non-equilibrium term reproduced the general trends
on the pressure difference, a satisfactory fit to the experimental data was not
obtained. After the maximum peak of the overshoot was reached, the simulated
pressure difference decreased more rapidly that the observed one. This might be
due to the functional form chosen for τ (Sw ). Also an inverse modelling approach
and parameter optimization may prove helpful.
In Chapter 8 hysteresis in capillary pressure is studied. Capillary pressuresaturation relationship is found to exhibit a hysteretic behavior when reversing
the process from drainage to imbibition. Hassanizadeh and Gray (1990) proposed that the capillary pressure-saturation relationship is a two-dimensional
projection of a more elaborate functional dependence and a third variable is
needed to explicitly define the state of the system. Thus, the hysteretic behavior can be modelled through the inclusion of specific interfacial area. Capillary
pressure and its functional relationship is investigated also at the micro-scale.
A series of experiments in a 2D micro-model were carried out to investigate the
role of specific interfacial area in modelling capillarity in two-phase flow. Under equilibrium condition, at each pressure step, external pressure and images
of fluid distributions were recorded. The fluids were allowed to flow into the
micro-model and reach the opposite side. Thus, two different flow conditions
could be distinguished: static and steady-state. With the term equilibrium we
refer to the state where fluid-fluid interfaces do not move and saturation remains
constant. The images are analyzed in order to extract average saturation, fluidfluid curvature, and the specific interfacial area. In these calculations, only the
interface spanning over the width of a pore is considered. We referred to this
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Chapter 9. Summary and Conclusions
as main terminal interface. Whereas the interface between wetting-nonwetting
phase film left in the corners of the pores were not considered. In order to establish whether the data points all fall in the same Pc − S − awn surface, the
ratio approach is used. The measured specific interfacial area is divided by the
specific interfacial area obtained from the interpolated surfaces. The closer the
ratio to unity is, the closer are the two surfaces. The results from the equilibrium
drainage and imbibition surfaces show an average ratios of 0.8 and 1.28 which
are in the range of the measurement error 10 − 15%. Thus, within measurement
error, drainage and imbibition surfaces under equilibrium can be considered as
the same surface. The results from non-equilibrium drainage and imbibition
experiments show that the ratios between measured and interpolated values of
specific interfacial area are 1.08 and 0.92. This also indicates that under nonequilibrium conditions, independent of the process (drainage or imbibition) the
data points fall on the same surface Pc − S − awn . The same ratio approach
is used to investigate whether the non-equilibrium and equilibrium data points
under drainage or imbibition conditions fall in the same surface. The average
ratio between measured specific interfacial awn during non-equilibrium drainage
to the interpolated values on the equilibrium drainage surface is 1.31. While
the ratio between the measured imbibition specific interfacial area under nonequilibrium to the interpolated ones under equilibrium imbibition gives 1.25.
These ratios are close to each other and within the measurement error the surfaces for drainage and imbibition under equilibrium and non-equilibrium are the
same. These results support the conjecture put forward by Hassanizadeh and
Gray (1993a), who suggested that the hysteretic in pc − S relationship may be
modelled through the inclusion of specific interfacial area in this relationship.
Samenvatting
De belangstelling voor het begrip van meerfasestroming in poreuze media en een
goede beschrijving daarvan is de laatste decennia enorm toegenomen, dankzij
het belang voor veel technische toepassingen zoals zuivering van verontreinigde
gebieden, de productie van papier en van brandstofcellen. In dit proefschrift
worden verschillende aspecten onderzocht met betrekking tot de theorie van
tweefasestroming in poreuze media.
Hoofdstuk twee geeft een overzicht van de grondbeginselen van de fysische
processen die de basis vormen voor de stroming van onmengbare vloeistoffen
(twee fase model) op microschaal en macroschaal.
Hoofdstuk drie focust op de definitie van gemiddelde druk. Hoewel men in
eerste instantie geneigd is te denken dat de definitie van gemiddelde vloeistofdruk eenvoudig is, is het antwoord op een simpele vraag zoals: Wat is de correcte
gemiddelde druk, niet simpel. Verschillende middelingsoperatoren worden geı̈ntroduceerd en gebruikt om gemiddelde vloeistofdruk te verkrijgen. De meest
gebruikte middelingsoperator is de intrinsic phase-volume average die een weging geeft van de waardes van de fasedruk en de faseverzadiging voor elk van de
verschillende fasen. Een alternatief is de simple phase-average die slechts een
gemiddelde druk geeft over het gebied waarin de fase daadwerkelijk aanwezig
is. Nog een alternatief is de simple-average, onder aanname dat de druk van
een fase α is gedefinieerd overal in het beschouwde gebied. Deze waarde geldt
als een gemiddelde over het hele gebied. De centroid-corrected phase average
druk of kortweg centroid-corrected pressure is geı̈ntroduceerd door Nordbotten
et al.(2008) als een onderdeel van een stelsel van macroschaal drukken. Deze
middelingsoperator corrigeert de intrinsic phase average druk voor afwijking van
de centroid van het middelingsvolume en de centroid van de fase.
We hebben de verschillen tussen deze middelingsoperatoren bestudeerd door
187
ze toe te passen op statische evenwichtssituaties in een verticaal homogeen
domein. We overwegen zowel statische als de dynamische situaties, waar een
non-wetting phase geı̈njecteerd wordt vanaf de bodemgrenslaag van het domein.
In statisch evenwicht, zijn druk en verzadigingsverdelingen bepaald voor een
gegeven overdruk ∆P van de non-wetting fase aan de grenslaag in de bodem
van het meetdomein. Deze verdeling is gebruikt in de berekening van de gemiddelde druk en verzadiging. Een belangrijk aspect van statisch evenwicht is dat de
totale phase potential (dat betekent druk plus zwaartekrachtinvloed) constant is
over het hele domein. Hierdoor zal dit gemiddelde gelijk zijn aan dezelfde constante. We hebben gevonden dat de gemiddelde druk volgens de intrinsic-phase
volume bepaling die algemeen gebruikt wordt in middelingsstudies resulteert in
een gradiënt in de total phase potential, dat betekent dat het hierboven genoemde criterium aanvechtbaar is. In feit voldoet alleen de centroid-corrected
operator aan dit criterium. Echter, bij hoge verzadigingen kan het gebruik van
het centroid-corrected gemiddelde aanleiding geven tot negatieve waarden voor
het verschil tussen de gemiddelde druk van de non-wetting en wetting fase. Voor
algemene drainage is het verschil in de gemiddelden operatoren minder belangrijk omdat beide fasen vanaf het begin aanwezig zijn op een zodanige manier
dat het verschil tussen de centroids van de fasen en het midden van het domein
betrekkelijk klein zijn. Na het definiëren van de gemiddelde druk rijst de vraag:
Wat is in feite de gemiddelde capillaire druk? Wat is de goede manier van
bepalen van capillaire druk op macro schaal? In deze studie wordt een serie
drainage en imbibitie experimenten uitgevoerd teneinde de relatie van capillaire
druk versus verzadiging onder evenwichtscondities te bepalen.
In hoofdstuk vier wordt een experimentele opstelling gepresenteerd inclusief
monstervoorbereiding, de experimentele procedure en de voorzieningen voor het
meten van calibratiekrommen en locale drukken en verzadigingen zoals de gemiddelde verzadiging over het hele domein.
In hoofdstuk vijf wordt een procedure voorgesteld om de capillaire druk
op te voeren. Algemeen gebruik bij het bepalen van capillaire druk is het
aftrekken van de wetting fase druk van de nonwetting fase druk zoals gemeten buiten de vloeistofreservoirs. Dit wordt dan aangenomen representatief
te zijn voor de gemiddelde capillaire druk voor de vloeistoffen in het poreuze
medium. Echter, de uitwendige vloeistofdrukverschillen corresponderen niet met
de capillaire druk-verzadigingskromme die representatief is voor de vloeistoffen
in het poreuze medium. In feite genereert deze methode een onnauwkeurigheid
door zwaartekrachteffectenwat zeker in een lange kolom een rol speelt. De
uitkomst van de normale procedure voor het meten van de capillaire drukverzadigingskromme is niet de echte capillaire druk en kan niet een poreus
188
medium met vloeistofstromen karakteriseren. In dit onderzoek is de gemiddelde capillaire druk bepaald, gebaseerd op de gemiddelde vloeistofdruk met
gebruikmaking van de hierboven aangegeven verschillende operatoren. Primaire
drainage, totale drainage en totale imbibitieexperimenten zijn uitgevoerd in een
relatief homogeen gepakte zandkolom onder evenwichtsomstandigheden. De gebruikte vloeistoffen zijn: PCE als non-wetting fase en gedemineraliseerd ontlucht
water als wetting-fase. In een drainageproces is de kolom initieel volledig waterverzadigd en is PCE van onderaf geı̈njecteerd. De injectiedruk is opgevoerd in kleine stappen en telkens 24 uur constant gehouden zodat een evenwicht
was bereikt voordat de volgende stap genomen werd. Gedurende imbibitie is
de stroomrichting omgekeerd. De druk van de injectievloeistof in debodemgrenslaag is verminderd in kleine stappen en van bovenaf is water in de kolom
gelaten bij een verwaarloosbare overdruk. De druk van de verschillende fasen
en de verzadiging binnenin de kolom zowel als de externe druk en gemiddelde
verzadiging zijn bepaald bij iedere evenwichtssituatie. Verschillende averaging
operators die hierboven geı̈ntroduceerd zijn, zijn gebruikt om de gemiddelde
druk van elke van de vloeistoffasen te bepalen, gebaseerd op de druk in vier
meetpunten. De gemiddelde capillaire druk die representatief is voor het totale gemiddelde domein is verkregen door het bepalen van het verschil van
de wetting druk en de nonwetting druk. De vergelijking tussen gemiddelde
capillaire druk-verzadiging krommen toont aan dat de kromme gebaseerd op
de centroid-corrected average beter overeenkomt met de referentiekromme ofwel
de potential-based Pc − Sw kromme, hoewel de potential-based kromme alleen
geldig is onder statische condities en de centroid-corrected average kan worden gebruikt voor stromingssituaties. Sterker nog, opgemerkt moet worden dat
binnen de meetfout de lokale primaire drainage capillaire druk kromme dicht
ligt bij de gemiddelde capillaire druk-verzadigingskromme kromme, gebaseerd
op de centroid-corrected average operator. Onder evenwichtsomstandigheden
en voor een homogeen poreus medium kan dus de lokale capillaire druk en de
gemiddelde capillaire druk-verzadigingskromme kromme beschreven worden met
dezelfde functionele relatie Pc (Sw ). Gedurende algemene drainage en algemene
imbibitie is het verschil tussen de gemiddelde fasedrukken gebaseerd op verschillende averaging operators verwaarloosbaar gebleken. Dit komt doordat beide
fasen van het begin af aanwezig waren en de centroids van de verschillende fasen
en de centroid van het gemiddelde domein dicht bij elkaar liggen.
In hoofdstuk zes zijn de niet-evenwichtseffecten van de drukken van de verschillende fasen onderzocht. Capillaire druk-verzadiging afhankelijkheid is een
eigenschap van het systeem vloeistof-medium en is bepaald onder evenwichtsomstandigheden. Hoewel dit normaal gebruikt wordt om processen te beschrijven
189
onder niet-evenwichtsomstandigheden. Kalaydjian (1992) en Hassanizadeh and
Gray (1990) gebaseerd op de thermodynamische benaderingswijze suggereren
zij dat onder niet-evenwichtsomstandigheden het verschil in vloeistofdrukken
gelijk zou moeten zijn aan de capillaire druk plus een extra term die proportioneel is met de snelheid van verandering van verzadiging. De hiermee
overeenkomende coëfficiënt τ wordt daarbij aangenomen een functie van de
verzadiging te zijn. Deze afhankelijkheid is een bron van discussie. Verschillende vragen komen op: Hoe kunnen drukverschillen en capillaire drukken met
elkaar vergeleken worden onder niet-evenwichtsomstandigheden? Wat is de functionele afhankelijkheid τ (Sw )? Is deze coëfficiënt afhankelijk van de fysieke
hoogte van het beschreven domein? Een serie dynamische primaire drainage
experimenten is uitgevoerd om de niet-evenwichtseffecten te bepalen van de
vloeistofverschildruk-verzadigingsafhankelijkheid. In dit geval is de druk ineens
opgevoerd tot een grote waarde en gedurende het experiment constant gehouden.
De injectiedrukken bedroegen 20 kPa, 30 kPa, 35 kPa en 38 kPa. De resulterende niet-evenwichtsverschildruk-verzadigingskrommen bleken hoger te zijn
dan de capillaire druk-verzadiging krommen. Sterker nog, de niet- evenwichtsdrukverschillen tonen een niet-monotoon gedrag met een overshoot die meer
uitgesproken is bij hogere injectiedruk. Gemiddelde vloeistofdrukverschillen en
verzadigingen zijn berekend voor het kolomdomein, gebaseerd op lokaal gemeten drukken en verzadigingen. Zoals reeds aangetoond is voor de steady-state,
is gebleken dat de intrinsieke average-operator het gemiddelde van de nonwetting fase overschat. Dit gemiddelde effect is zeer uitgesproken in het nietevenwichtsgeval als de toegepaste drukken veel hoger zijn. Een gemiddeld nietevenwichtsdrukverschil dat bepaald is met de centroid-corrected average operator
ligt nog altijd boven de gemiddelde capillaire druk. Bovendien is ook een nietmonotoon gedrag bepaald voor een hogere injectiedruk. De niet- evenwichtscoëfficiënt τ op de lokale punten is bepaald met drie verschillende methodes. Hoewel
elke methode een verschillende trend in de variatie met de verzadiging geeft,
blijft de τ -waarde op deze lokale schaal in de orde van 105 Pa.s. De nietevenwichtscoëfficiënt is op deze manier geschat op de kolomschaal door toepassing van gemiddelde verzadiging in gemiddelde fasedruk. De resultaten tonen
aan dat deze coëfficiënt een orde van grootte hoger ligt dan de lokale dempingscoëfficiënt, ongeveer 106 Pa.s. Het valt op dat deze zelfs nog groter is als
de intrinsic-phase-average wordt gebruikt. Hoewel het moet worden benadrukt
dat dit niet overeenkomt met het kwadraat van de lengte van het gemiddelde
domein, zoals voorgesteld door Dahle et. al (2005) voor een bundel buizen.
In hoofdstuk zeven worden niet-evenwichts primaire drainage experimenten
gesimuleerd in het laboratorium door het toepassen van de traditionele tweefasen
190
stromingsvergelijkingen waarna hieraan toegevoegd wordt de niet-evenwichts
term zoals voorgesteld door Kalaydjian (1992) en Hassanizadeh and Gray (1990).
Verschillende functionele afhankelijkheden van de niet-evenwichtscoëfficiënt τ
op de wetting-fase verzadiging, namelijk lineair, kwadratisch, error-function en
exponentieel zijn overwogen. Teneinde de gesimuleerde resultaten te kunnen
vergelijken met experimentele data zijn de non-wetting fase en wetting fase
drukken zowel als de wetting verzadiging gemiddeld over een gebied vergelijkbaar met het meetgebied van de druk-sensoren en de verzadigingssensoren
(TDR) sensoren. De resultaten tonen aan dat, onder niet-evenwichtscondities
de gekozen averaging operator voor het bepalen van de gemiddelde fase-druk een
belangrijke rol speelt, zelfs op de schaal van de sensor. In feite zijn grote verschillen in de druk aan de grenslaag verkregen als de intrinsic-phase averaging of
de centroid-corrected toegepast is. Het is gebleken dat voor een injectiedruk van
20 kPa de niet-evenwichtseffecten op de vloeistofdrukverschillen niet significant
zijn. Voor kleine waarden van τ verschillen de gesimuleerde resultaten niet van
die verkregen zonder de niet-evenwichtsterm. De experimentele data bij hoge
injectiedruk van 30 kPa tot 35 kPa and 38 kPa vertoont een grote overshoot van
de niet-evenwichtsdrukverschillen in de tijd. Numerieke resultaten tonen dat
dit niet-monotone gedrag slechts kan worden gereproduceerd als de dynamische
term meegenomen wordt in de vergelijkingen die de tweefasenstroming beschrijven. Hoewel de niet-evenwichtsterm de algemene trend in drukverschillen volgt,
is niet een bevredigende overeenkomst met de experimentele data verkregen. Nadat het maximum van de overshoot is bereikt, valt het gesimuleerde drukverschil
sneller terug dan de waargenomen waarde. Dit kan komen door de keuze voor de
functionele formule voor τ (Sw ). Ook een omgekeerde modellerings benadering
en optimalisering van de parameters kan waardevol blijken.
In hoofdstuk acht is de hysteresis van de capillaire druk bestudeerd. De
afhankelijkheid van de capillaire druk versus verzadiging blijkt een hysterese
te vertonen als het proces van drainage omgekeerd wordt in imbibitie. Hassanizadeh and Gray (1990) heeft voorgesteld dat de capillaire druk-verzadiging
afhankelijkheid een tweedimensionale projectie is van een meer nauwkeurige
functionele afhankelijkheid en dat een derde variabele nodig is om de toestand
van het systeem volledig te beschrijven. Dus kan het hysterese gedrag gemodelleerd worden door het toevoegen van een specific interfacial area term. De
functionele afhankelijkheid van capillaire druk is ook op microschaal onderzocht.
Een serie experimenten in een 2D micromodel is uitgevoerd om de specifieke
rol van interfacial area te onderzoeken in het modelleren van capillariteit in
tweefasenstroming. Onder evenwichtsomstandigheden zijn de uitwendige druk
en camerabeelden van de vloeistofverdelingen bij iedere opvolgende stap in de
191
drukwaarde opgeslagen. De vloeistoffen werden in het micromodel gebracht tot
dit verzadigd was. Op deze manier zijn twee verschillende stromingscondities onderscheiden: statisch en steady-state. Met de term evenwicht refereren we aan de
situatie waarin vloeistof-vloeistof grenslagen niet bewegen en de verzadiging constant blijft. De beelden zijn geanalyseerd om gegevens te verkrijgen van gemiddelde verzadiging, de vorm van het vloeistof-vloeistof contact en het specifieke
oppervlak hiervan. In deze berekeningen is alleen naar de grenslaag-spanning
over de breedte van de porie gekeken. We refereren hieraan als algemene terminal interface. Het interface tussen de wetting fase en de nonwetting fase in
hoeken van de poriën is niet meegenomen.
Teneinde te bepalen of de datapunten in hetzelfde oppervlak vallen als de
Pc − Sw − awn oppervlak is de ratiobenadering gebruikt. De gemeten specifieke
interface oppervlakte is gedeeld door de specifieke oppervlak verkregen uit de
geı̈nterpoleerde oppervlakken. Hoe dichter de verhouding bij 1 komt hoe beter
de twee oppervlakken samenvallen. Het resultaat van de evenwichtsdrainage enimbibitie oppervlakken toont een gemiddelde verhouding van 0,8 en 1,28 die binnen de meetfout van 10% tot 15% ligt. De drainage en imbibitie oppervlakken
onder evenwichtsomstandigheden kunnen dus binnen de meerfout geacht worden hetzelfde oppervlak te hebben. De resultaten van niet-evenwicht drainage
en imbibitie experimenten tonen aan dat de verhouding tussen de gemeten en de
geı̈nterpoleerde waardes voor de specifieke interface oppervlakken 1,08 en 0,92
zijn. Dit geeft aan dat ook bij niet-evenwichtsomstandigheden, onafhankelijjk
van het proces (drainage of imbibitie) de meetpunten op hetzelfde Pc − Sw − awn
oppervlak vallen. Dezelfde verhoudingsbenadering is gebruikt om te bepalen of
de niet-evenwichts- en evenwichtsmeetpunten onder drainage of imbibitie omstandigheden in hetzelfde oppervlak vallen. De gemiddelde verhouding tussen
gemeten specifieke interface awn gedurende niet-evenwichtsdrainage ten opzichte
van de geı̈nterpoleerde waardes bij evenwichtsdrainage is 1,31, terwijl de verhouding tussen de gemeten specifieke interface oppervlakte bij imbibitie onder
niet-evenwicht ten opzichte van de geı̈nterpoleerde waarde onder evenwicht 1,25
geeft. Deze ratio’s liggen dicht bij elkaar en binnen de meetfout van de oppervlakte voor drainage en imbibitie (onder evenwicht en niet-evenwicht) zijn
ze gelijk te stellen. Deze resultaat-ondersteunende veronderstelling van Hassanizadeh and Gray (1993a) suggereert dat de hysterese in de Pc − Sw relatie
gemodelleerd kan worden door hierin de specifieke interface oppervlak te betrekken.
192
Appendix: Determination
of the derivatives of the
intrinsic phase-volume
average pressure and
position vector
In this appendix, we show the derivation of the derivative of the intrinsic phasevolume average pressure. First, the definition of the intrinsic phase-volume averaged pressure (Equation(2.1)) is inserted into the derivative, assuming constant
porosity:
∇ < Pα >a = ∇
R
V
1
ǫSα dV
Z
ǫSα Pα dV
V
=∇
1
< Sα > ǫV
Z
ǫSα Pα dV
(9.1)
By applying the product rule to Equation 9.1, and assuming constant porosity, we get:
Z
Z
∇ < Sα >
1
∇ < Pα >a = −
S
P
dV
+
∇
S
P
dV
(9.2)
α α
α α
Sα2 V
< Sα > V
V
V
The averaging theorem of Slattery (1967)states that:
193
1
∇ < ω >= ∇ <
V
Z
1
ωdV >=
V
Z
(9.3)
nωdV
where ω denotes a function, V is the averaging volume, n is the unit normal
vector along the boundary of volume V, denoted by ∂V . Thus, the derivative
of an average function ω can be expressed in terms of a surface integral. Using
this theorem, Equation 9.2 becomes:
R
nSα dA
∇ < Pα > = −
< Sα > 2 V 2
a
∂V
Z
Sα Pα dV +
1
Z
< Sα > V
nSα Pα dA
(9.4)
∂V
The first term can be rewritten by applying the definition of the intrinsicphase-volume-average pressure:
Z
Z
1
a
a
∇ < Pα > =
− < Pα >
nSα dA +
nSα Pα dA
(9.5)
< Sα > V
∂V
∂V
A similar approach can be followed in the derivation of the derivative of the
intrinsic-phase-volume-average position vector z, resulting in:
∇· < z >a =
1
< Sα > V
− < z >a ·
Z
∂V
n · Sα dA +
Z
∂V
n · Sα zdA
(9.6)
For the particular situation of a vertical domain and flow from the bottom
of the domain to the top, Equation 9.5 and 9.6 become (Nordbotten, personal
communication):
∂
1
< Pα >a ≡
− < Pα >α Sαtop − Sαbot + Sαtop Pαtop − Sαbot Pαbot
∂z
< Sα > H
(9.7)
where H is the height of the modelling domain. The superscripts ’top’ and
’bot’ refer to the value of the variable (Sα , Pα , and zα ) at the top and the bottom
of the domain, respectively.
∂
1
< zα >a ≡
− < zα >a Sαtop − Sαbot + Sαtop zαtop − Sαbot zαbot
∂z
< Sα > H
(9.8)
194
Bibliography
Slattery, J. (1967), Flow of viscoelastic fluid through porous media, A.I.Ch.E.J,
13.
195
Acknowledgments
I would like to thank a number of people that have helped and supported me to
complete my doctoral thesis.
First of all I would like to express all my sincere gratitude to my supervisor
Majid Hassanizadeh for introducing me in the intricate world of multiphase in
porous media. Maijd, I have learned a lot from you in these past years. I am
very grateful I had you as my supervisor. Your way of looking at things, your
criticism and rigor together with your contagious enthusiasm have shown me the
way research should be done. I appreciate the time you devote to my manuscript
(I hope you have improved your Italian though). Your door has been always open
and not only for scientific-related matters. Thank you for your encouragements
and guidance.
To Pieter Kleingeld. It would not have been a successful experimental work
without your precious advices. Developing the experimental set-up was not a
straightforward task, several issues came across that required many changes to
the original plan. Thank you for being enthusiastic and willing to help. Thank
you also for helping me with the Dutch translation. I wish to thank Timo
Heimovaara for the many advices in writing the algorithm for the TDR data
acquisition, Adam Bezuijen and Jack van der Vegt for their advice on the experimental work when the set-up was still in GeoDelft. A special gratitude to
Laura Pyrak-Nolte from Purdue University. Visiting your group in Purdue was
a pleasure. I truly enjoyed working with you, our weekly meetings, and the
many discussions related to micro-models. I really appreciate your comments
and suggestions that helped improving this manuscript. Likewise, to Mart Oostrom and his group at Pacific Northwest National Laboratory, for the fruitful
discussions on multiphase flow processes and the nice time spent in Richland.
To Cas Berentsen for his guidance on the numerical simulations and many nice
197
discussions for which I’m deeply grateful. To Suze-Anne Korteland for the fruitful discussions on the averaging operators. It was a pleasure working with you.
And to Rainer Helmig from Stuttgart University for the all discussions about
non-equilibrium effects and averaging operators during the NUPUS meetings
and workshops.
In the past years the Environmental Hydrogeology Group became like a second family for which I would like to thank all the members for making these
years unforgettable. First of all, I should like to thank Margreet for the nice
chats during coffee breaks, for reminding me about lunch time, and for helping
me with the administrative work. Thank you Ruud for your enthusiasm, for the
delicious dinners you prepared for all of us at your place, for showing us the
beauty of your city, Rotterdam, and for the great rock concerts. Many thanks
off course to you Majid for entertaining us with your anecdotes and jokes. Many
thanks to my dear colleagues for making the time at the university enjoyable:
Marian, Henk, Cas, Mariene, Bert-Rik, Sam, Phil, Saeed, Vahid, Amir, Reza,
Nikos, Brijesh and Eric.
I wish to thank all the wonderful people I meet in the Netherlands in the last
years that indirectly contribute to this work. Some words go to my friend and
running mate Mariene. I will never forget our never ending chats. I shared with
you really a good time. But, especially I would like to thank you for sharing also
the less nice time and never hesitating to give me your support whenever it was
needed. To Roberta, I still remember your first e-mail looking for other Italians.
It was funny. Thank you for listening and for being always around whenever
I needed a friend. To Ueli, for your optimism and impeccable music taste. To
Thomas for being a good listener, for offering your support and for the nice
climbing times in Amsterdam together with Catherine. Catherine thank you
for keeping me up to date on the theater performances, and the delicious and
unforgettable dinners. To Cornelia for the nice time together, for the evenings
spent at the cinema to watch movies that sometimes only you, the director and
I would appreciate. To Siska for our nice girly time. Max, thank you for the
unforgettable time in Amsterdam. Pierre, thank you for your wise advices and
the great walking times at the dunes. I should also like to thank Aco, Joost,
Xavier, Christian and Cöme for the party times. To Fred for the delicious cakes.
A special thanks goes to my wonderful friends from Sardinia. Dear Angela
thank you for being always by my side: t.v.b.. Alessandro, Marcella, Alberto
thank you for looking after me. Un abbraccio fortissimo a tutti!
A special thanks to Alessandro for the many suggestions on the layout of this
thesis, but, especially for making me laugh even in the most stressful moments,
for your patience and for your encouragement and support: un super bacio!
198
Vorrei ringraziare I miei genitori. Marta e Francesco per il loro amore incondizionato e continuo sostegno in tutte le vie che ho intrapreso. Vorrei ringraziare
i miei fratelli Andrea, e Marco e la sua amata Valentina. Un grazie va al mio
dolcissimo nipotino Leonardo per avere portato in casa una ventata di allegria
e spensieratezza. Ringrazio inoltre le mie zie: Luciana, Vittorina e Eugenia che
mi sono state sempre vicine.
Simona
199
About the Author
Simona Bottero was born in Cagliari, Sardinia, Italy. She received her M.Sc
Cum Laude from Cagliary University, Faculty of Earth Science with final thesis in Hygrogeology. Her thesis focused on hydrogeological water balance and
flow mechanisms in the sub-surface. In 2002 she was awarded by the European
Union with a Leonardo da Vinci Grant that allowed her to spend four months
at the Research and Developments Department of Amsterdam Water Supply
(now called Waternet), in the Netherlands. After those months, she worked
in the same department as geohydrologist where she dealt with transport of
thermal energy in artificial recharge area and computation of groundwater flow
velocity by simulating the transport of thermal energy. In 2003 she worked as
Hydrogeologist at TNO, National Research Institute of the Netherlands, at the
Groundwater Department in Utrecht. Her work involved in-situ groundwater
temperature data acquisition, data analysis and modeling groundwater flow and
heat transport processes. In July 2004 she took a PhD position at the Environmental Hydrogelogy Group, Utrecht University, to work on two-phase flow
processes in porous media. The results of her research are summarized in this
PhD thesis. Simona now works as post-doc researcher at the Environmental
Biotechnology Group, Delft University of Technology.
201
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