DissHansGraf

DissHansGraf
DISSERTATION
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the Rupertus Carola University of
Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
presented by
Diplom-Physiker Hans Graf
born in Öhringen
Oral examination: April 21st, 2004
Experimental Investigations on
Multiphase Phenomena in Porous
Media
Referees:
Prof. Dr. Kurt Roth
Prof. Dr. Peter Bastian
Experimentelle Untersuchungen von Mehrphasenphänomenen
in porösen Medien
Die Beschreibung des Transports von Wasser und gelösten Stoffen in porösen Medien
basiert traditionell ausschließlich auf den hydraulischen Eigenschaften. Da für den ungesättigten Fall aber Wasser und Luft gleichzeitig denselben Porenraum teilen, ist eine
genauere Untersuchung der Gasphase notwendig, um ihren Einfluss auf die grundlegenden
Prozesse (Phasenkontinuität, Hysterese, Phaseneinschluss,. . . ) speziell nahe Wassersättigung zu verstehen. Hierfür wurde eine Apparatur zur experimentellen Bestimmung der
hydraulischen Eigenschaften mittels eines transienten Experiments dahingehend erweitert,
dass simultan die Bestimmung der wichtigsten Gasflussparameter möglich war. So konnten
Kontinuität und Leitfähigkeit der Gasphase an Laborsäulen bestimmt werden. Die Auswertung der hydraulischen Daten erfolgte mittels inverser Modellierung. Darauf basierend
wurden die pneumatischen Eigenschaften mit verschiedenen Gasleitfähigkeitsmodellen simuliert.
Die Möglichkeiten der simultanen Bestimmung hydraulischer und pneumatischer Eigenschaften wurden an künstlichen porösen Medien aus gesintertem Glas demonstriert.
Ein Vergleich der Messungen mit Luftleitfähigkeitsmodellen hat die Notwendigkeit einer
Reskalierung der effektiven Luftsättigung zur Berechnung der Leitfähigkeit gezeigt.
Anhand verschiedener homogen und heterogen geschütteten Sandpackungen konnte der
Einfluss der Struktur auf die hydraulischen und pneumatischen Materialeigenschaften verdeutlicht werden. Zwei pathologische Strukturen wurden rein hydraulisch und zusätzlich
auch noch pneumatisch analysiert. Die Kombination beider Messmethoden ermöglichte
Rückschlüsse auf die grundlegenden Strukturelemente der Proben.
Experimental Investigations on Multiphase Phenomena
in Porous Media
Modeling of water flow and solute transport in porous media is typically based on the water
dynamics only, while the gaseous phase is neglected. Since the two fluids share the same
pore space a particular investigation of the gaseous phase is mandatory to understand its
influence on the basic processes (continuity, hysteresis, entrapment,. . . ) especially near
water saturation.
For the multiphase measurements an existing multistep outflow setup for determination
of hydraulic properties with laboratory sized columns, was improved by an additional airflow measurement device where gas phase continuity and conductivity could be measured
simultaneously. Measured hydraulic data was analyzed by inverse modeling on which the
pneumatic data analysis was based. Several gas conductivity models were tested.
The possibilities of a combined measurement of hydraulic and pneumatic properties
were demonstrated with artificial porous media made of sintered glass. The comparison
of measurement and simulations of air conductivity showed the necessity of a rescaling of
the effective air saturation for predictions.
The influence of the sample structure on the hydraulic and pneumatic properties was
illustrated with several homogenous and heterogeneous samples of repacked sands. The
differences between purely hydraulic and combined measurement could be shown with experiments carried out with two pathologic structures. For this samples the basic structure
elements could be detected by the combination of hydraulic and pneumatic measurements.
Contents
Zusammenfassung
i
Summary
i
List of Figures
vii
List of Tables
ix
List of Symbols
xi
1 Introduction
1
2 Theory of Immiscible Fluids in Porous Media
2.1 Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 REV, Averaging, and Continuum Approach . . . . . . . . .
2.1.2 Porosity, Liquid Content and Saturation . . . . . . . . . . .
2.2 Capillarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Micro Scale Effects . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Capillarity Effects at the Pore Scale . . . . . . . . . . . . . .
2.2.3 Capillarity Effects at the Continuum Scale . . . . . . . . . .
2.2.4 Hysteresis in Soil Water Retention Function . . . . . . . . .
2.2.5 Experimental Methods to Determine Pressure-Saturation Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.6 Empirical Descriptions for Pressure-Saturation Relation . . .
2.2.6.1 The Brooks and Corey Model . . . . . . . . . . . .
2.2.6.2 The van Genuchten Model . . . . . . . . . . . . . .
2.2.6.3 Pressure-Saturation Hysteresis Models . . . . . . .
2.3 Fluid Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Conductivity at the Continuum Scale . . . . . . . . . . . . .
2.3.2 Experimental Methods of Determining Unsaturated Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Parameterization of Unsaturated Conductivity . . . . . . . .
2.3.3.1 Purely Empirical Conductivity Models . . . . . . .
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iii
Contents
2.4
2.5
2.3.3.2 Geometry Based Conductivity Models . . . . .
2.3.4 Effects Influencing Air Conductivity . . . . . . . . . . . .
2.3.5 Hysteresis Effects in Conductivity Functions . . . . . . .
Parameterizations of Hydraulic and Pneumatic p-k -S –Relations
2.4.1 The van Genuchten-Mualem Model — VGM . . . . . . .
2.4.2 The van Genuchten-Burdine Model — VGB . . . . . . .
2.4.3 The Emergence-Point Model — EP . . . . . . . . . . . .
Dynamic Water Transport . . . . . . . . . . . . . . . . . . . . .
3 Material and Methods
3.1 Multistep Outflow Method . . . . . . . . . . . . .
3.2 Experimental Setup for Multiphase Measurements
3.2.1 Mulitstep Outflow Measurement Device . .
3.2.2 Mini-Tensiometers . . . . . . . . . . . . .
3.2.3 Air-Flow Measurement Device . . . . . . .
3.3 Preparation of the Samples . . . . . . . . . . . . .
3.4 Experimental Settings . . . . . . . . . . . . . . .
3.5 Estimation of Hydraulic Parameters . . . . . . . .
3.6 Calculation of Air Conductivity . . . . . . . . . .
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4 Experimental Investigations with Artificial Porous Media
4.1 Homogeneous Sintered Glass Columns . . . . . . . . .
4.1.1 Homogeneous Glass Column – P250 . . . . . . .
4.1.2 Homogeneous Glass Column – P100 . . . . . . .
4.2 Structured Column . . . . . . . . . . . . . . . . . . . .
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Two Phase Flow Measurement with repacked Sand Columns
5.1 Characterization of Homogeneous Sands . . . . . . . . . .
5.1.1 Coarse Sand – S-I . . . . . . . . . . . . . . . . . . .
5.1.2 Fine Sand – S-II . . . . . . . . . . . . . . . . . . .
5.2 Structured Sand Columns . . . . . . . . . . . . . . . . . .
5.2.1 Vertically Structured Sand Packing – S-H-I . . . . .
5.2.2 Horizontally Structured Sand Packing – S-H-II . . .
5.3 Results of Measurement with Sand
Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Conclusions
109
Bibliography
113
iv
Contents
Appendix
123
A Setup
124
A.1 Pressure Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A.2 Experimental problems . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B Sintered glass
127
B.1 Chemical composition . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.2 Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.3 Influence of Sample Structure on Transport Processes . . . . . . . . . 128
v
Contents
vi
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.1
Typical length scales in porous media . . . . . . . . . . . . . . . . . .
Definition of a representative elementary volume . . . . . . . . . . . .
Surface tension and contact angle . . . . . . . . . . . . . . . . . . . .
Surface element in a capillary . . . . . . . . . . . . . . . . . . . . . .
Model of pore cross-section with varying saturation of water . . . . .
Raindrop effect demonstrating contact angle for drainage and imbibition
Model for capillary hysteresis . . . . . . . . . . . . . . . . . . . . . .
Snap-off effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
By-Passing effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hysteresis in the pc (θ) relationship . . . . . . . . . . . . . . . . . . .
Comparison of the Brooks and Corey and van Genuchten retention
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dependent domain theory for hysteretic retention function . . . . . .
Geometric effects influencing permeability . . . . . . . . . . . . . . .
Schematic representation of retention curve and relative conductivity
functions for the wetting and the non-wetting phases during drainage
Dependent domain theory for hysteretic conductivity function . . . .
Comparison of different air conductivity models . . . . . . . . . . . .
Multistep outflow measurement curve . . . . . . . . . . . . . . . . .
Experimental setup for simultaneous measurement of water and air
dynamics in laboratory samples . . . . . . . . . . . . . . . . . . . .
External tensiometer designed for use with rigid porous media . . .
Porous plate to separate air and water at the lower boundary . . . .
Additional separator in air-flow channel . . . . . . . . . . . . . . . .
Example of an air-flow measurement . . . . . . . . . . . . . . . . .
Falling head permeameter . . . . . . . . . . . . . . . . . . . . . . .
Graphical user interface for control and evaluation of multistep outflow experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of data evaluation of combined multistep outflow and airflow measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
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. 65
Structured porous sintered glass medium . . . . . . . . . . . . . . . . 66
vii
List of Figures
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
X-ray tomograms of the sintered glass media . . . . . . . . . . . . .
Measurement and simulations of hysteretic outflow curve of P250
glass column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hydraulic functions of homogeneous glass columns . . . . . . . . . .
Measurement and simulation of hysteretic outflow curve of P100 glass
column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Structured sintered glass column . . . . . . . . . . . . . . . . . . . .
Measurement and simulations of hysteretic outflow curve of structured glass column . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of different air conductivity models and data measured
in heterogeneous glass column . . . . . . . . . . . . . . . . . . . . .
Comparison of simulated and measured air conductivity for structured glass column . . . . . . . . . . . . . . . . . . . . . . . . . . .
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. 82
Measurement and simulations of hysteretic outflow curve of sand S-I .
Hydraulic functions of homogeneous sands . . . . . . . . . . . . . . .
Multistep outflow with simultaneous air-flow measurement for sand
S-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulations of measured data in sand S-II . . . . . . . . . . . . . . .
Comparison of different air conductivity models and data measured
in homogeneous column S-II . . . . . . . . . . . . . . . . . . . . . . .
Simulation of air conductivity in homogeneous column S-II . . . . . .
Hysteresis in hydraulic functions of homogeneous S-II sand . . . . . .
Structured sand packing S-H-I . . . . . . . . . . . . . . . . . . . . . .
Multistep outflow measurement results for structured sand S-H-I . .
Comparison of different air conductivity models and data measured
in heterogeneous column S-H-I . . . . . . . . . . . . . . . . . . . . . .
Structured sand packing S-H-II . . . . . . . . . . . . . . . . . . . . .
Multistep outflow and air-flow measurement for sand structure S-H-II
Independent evaluation of multistep outflow and air-flow measurement for sand structure S-H-II . . . . . . . . . . . . . . . . . . . . .
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102
103
106
B.1 Simulation of water transport with low flow-rate for the heterogeneous
sintered glass column . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B.2 Simulation of water transport with high flow-rate for the heterogeneous sintered glass column . . . . . . . . . . . . . . . . . . . . . . . 130
viii
List of Tables
2.1
2.2
Values of the scaling factors for primary and main curves . . . . . . . 18
Selected empirical models for non-wetting phase conductivity . . . . . 32
3.1
Separator materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1
4.2
4.3
4.4
Material properties of the sintered glass columns . . . . . . . . .
Van Genuchten parameters for homogeneous glass sample P250
Van Genuchten parameters for homogeneous P100 glass sample
Van Genuchten parameters for heterogeneous glass sample . . .
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72
75
80
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
Material properties of the homogeneous sand columns . . . . . . . . .
Van Genuchten parameters for homogeneous sand sample S-I . . . . .
Van Genuchten parameters for homogeneous S-II sample . . . . . . .
Van Genuchten parameters for scanning cycles in S-II sample . . . . .
Material properties of the heterogeneous sand columns . . . . . . . .
Van Genuchten parameters for heterogeneous S-H-I sample . . . . . .
Van Genuchten parameters for heterogeneous S-H-II sample . . . . .
Independent van Genuchten parameters for heterogeneous S-H-II sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ix
List of Tables
x
List of Symbols
This list contains the most important symbols used. Where possible, a reference
(equation and page number) is given and the dimension is indicated in brackets.
The mathematical structure of symbols is indicated by their typographical appearance:
a scalar
a, b
a vector, unit length vector
A tensor
sin standard function
Subscripts usually refer to a component of a vector (x, y, z, or 1, 2, 3) or to the
phase (a, w, s).
The arguments of functions are suppressed if they are clear from the context,
e.g., ∂x f instead of ∂x f (x). They are written, however, if the dependence on an
argument is emphasized, e.g., Kd (Cw ) for a nonlinear adsorption isotherm. Similarly,
derivatives are written as shorthand operators: ∂x , ∂xx for the first and second partial
derivative with respect to x, respectively, and dt for the total derivative ∂t + v · ∇
∂¦
with respect to time. If the derivative is to be emphasized, the long form ∂x
is used.
Sign Convention
The normal vector n on the surface of some volume points outwards. The vertical
(z) axis points downward, in the direction of the acceleration of gravity. Its origin
is typically chosen at the soil surface. Accordingly, z is called the depth.
Lowercase Latin Symbols
b
h
ja
jw
ka
ka∗
Klinkenberg parameter
hydraulic head [L]
volumetric air flux [L T−1 ]
volumetric water flux [L T−1 ]
pneumatic conductivity [L T−1 ]
slip enhanced pneumatic permeability [L2 ]
2.56
2.15
3.1
2.32
2.42
2.56
37
21
52
27
34
37
List of Symbols
ka,c
ka,n
ki
kr,a
kr,w
kw
m
n
pa
pc
pe
pw
r
pneumatic conductivity of entire column [L T−1 ]
pneumatic conductivity of a single node [L T−1 ]
intrinsic permeability [L2 ]
relative pneumatic conductivity [-]
relative hydraulic permeability [-]
hydraulic conductivity [L T−1 ]
van Genuchten parameter
van Genuchten parameter
pressure in air phase [F L−2 ]
capillary pressure [F L−2 ]
entry pressure [F L−2 ]
pressure in water phase [F L−2 ]
pore radius [L]
3.9
3.9
2.33
2.46
2.45
2.37
2.15
2.15
2.6
2.6
2.12
2.6
2.7
63
63
28
35
35
30
21
21
11
11
20
11
12
2.67
2.32
2.25
2.39
2.2
2.64
2.60
2.12
2.57
2.12
2.1
2.12
2.12
43
27
25
34
9
42
39
20
39
20
9
20
20
2.15
2.5
2.70
2.33
21
10
44
28
Uppercase Latin Symbols
Ka,sat
Kw,sat
Pd
R
Sa
Sa,e
∗
Sa,f
Sa,r
Sa,t
Se
Sw
Sw,r
Sw,s
saturated pneumatic conductivity [LT −1 ]
saturated hydraulic conductivity [LT −1 ]
domain dependent factor [-]
mean pore radius [L]
air phase saturation [-]
effective air phase saturation [-]
effective mobile air phase saturation [-]
residual air saturation [-]
effective trapped air phase saturation [-]
effective water saturation [-]
water saturation [-]
residual water saturation [-]
maximum water saturation [-]
Lowercase Greek Symbols
α
γ
ηa
ηw
xii
van Genuchten parameter [L−1 ]
contact angle [ ° ]
viscosity of air [F T L−2 ]
viscosity of water [F T L−2 ]
List of Symbols
θa
volumetric air content [-]
2.2 9
θa,ea air content at emergence point of air phase [-]
2.64 42
θa,r residual air content [-]
2.38 34
θa,s maximum air content [-]
2.38 34
θw
volumetric water content [-]
2.1 9
θw,ew water content at emergence point of water phase [-]
2.63 42
θw,r residual water content [-]
2.1 9
θw,s saturated water content [-]
2.38 34
θ∆
saturation at reversal point [-]
2.25 25
λ
Brooks and Corey parameter
2.12 20
ρa
mass density of air [M L−3 ]
2.70 44
−3
ρw
mass density of water [M L ]
2.33 28
−2
σas
surface tension of air with respect to solid [M T ]
2.5 10
σwa surface tension of water with respect to air [M T−2 ]
2.5 10
−2
σws surface tension of water with respect to solid [M T ]
2.5 10
τ
tortuosity parameter
2.41 34
φ
porosity [-]
2.1 9
−1 −2
ψg
gravitational potential [ML T ]
2.8 12
−1 −2
ψm
matric potential [ML T ]
2.9 12
ψs
osmotic potential [ML−1 T−2 ]
2.9 12
−1 −2
ψtp
tensiometer potential [ML T ]
2.9 12
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xiii
Measurement of air conductivity as a function of water content is
a promising tool for obtaining new information useful in analyzing
problems of aeration as well as infiltration and drainage in soils.
However it is a tool which soil scientists seldom use and has never
been developed for routine use.
(A. T. Corey, 1986)
1 Introduction
Investigation of multiphase flow in porous media is important in many branches of
science and engineering. In the technical field, this includes blast furnaces, catalytic
converters, or fuel cells like the low temperature direct methanol fuel cells where
methanol, hydrogen and oxygen share the pore space of the reforming membrane.
In the field of bio reactors, the porous medium is the substrate for bacteria which
have to be ”fed” by fluids and gases, transforming them in their specific ways.
Historically, the petroleum industry was one of the earliest interested in a detailed
description of multiphase phenomena. Crude oil, natural gas, and water share the
pore space in the deposits and a distinct knowledge of the basic processes could
directly increase the efficiency of exploitation.
Huge fields of multiphase research are the agronomic and environmental sciences.
These include groundwater hydrology and soil science with their distinct branches.
The dynamics of water movement in soil is of great importance for many issues of
environmental and economic interest, for instance, for the quantity and quality of
groundwater, for agricultural production, or for the design and management of hazardous waste sites. The governing equations which describe the dynamics of water
movement are highly nonlinear due to the strong increase of soil’s hydraulic conductivity with water saturation. The problem arises most drastically with infiltration of
water, the phenomenon of greatest interest for the transport of dissolved chemicals
(fertilizers, pesticides, heavy metals, gasoline additives, . . . ) where an important
characteristic is the residence time of the solute in the highly reactive root zone.
Infiltration leads to the movement of roughly the same volumes of water and air in
opposite directions. For this reason, the system must be understood quantitatively
well near water saturation.
The dynamics of soil air has been of secondary interest to soil and agricultural
scientists, because the gas exchange between soil and the atmosphere was thought
to be unimportant for most phenomena. The attention was renewed, when the pollution of soil by organic compounds had to be removed. The pollutants, which came
from accidental spills or leakage in production plants or waste deposits, infiltrated
into the unsaturated zone where they would threaten agricultural acreages or the
groundwater. As many of these chemicals are only slightly soluble in water but
highly volatile they could be transported significantly by advective components of
the air phase, therefore, an understanding of the air phase as well as of the water
phase has become important.
1
1 Introduction
Motivation
The usual approach for description of unsaturated fluid flow involves the assumption
that the air phase essentially remains at a constant pressure, equal to atmospheric;
the system is then reduced to the consideration of the water phase only. This approach is called the Richards approximation (Richards 1931) and is the standard
method in soil science. Combined with some parameterizations for the material
properties, specifically for the soil water characteristic and for the hydraulic conductivity (e.g., Brooks and Corey 1966, van Genuchten 1980, Burdine 1953, Mualem
1976) it is possible to solve the resulting system of equations numerically. It has
been shown to be a good approximation for most applications, however, there are
some situations where air phase can significantly retard the movement of the water
phase.
Obviously, there exists an important domain near complete water saturation where
the assumptions of the Richards equation are violated and where a complete multiphase formulation is mandatory. While this insight is not new, its implications have
not often been appreciated. They affect the simulation of water movement near
saturation as well as the popular inverse methods (Kool et al. 1985, van Dam et al.
1994) for estimating hydraulic properties. The latter affect appears particularly severe, since, incorrectly prescribing the Richards equation as the dynamics model, can
fold all structural deviations into the estimated parameter functions. As mentioned
above in a more extended perspective, the problem raised is also relevant to other
issues, for instance for the aeration of soil and for the movement of non-aqueous
phase liquids.
Since different multiphase modeling approaches have been developed during the
last decades there is an need for data to support mathematical descriptions of consistent parametric two phase flow models.
In this study, a versatile experimental approach for studying water and air movement in nearly water-saturated porous media is demonstrated. For determination
of hydraulic material properties the transient method of multistep outflow is combined with an inverse parameter estimation approach. The aim of the experimental
design, is the simultaneous measurement of air conductivity within a running multistep outflow experiment. The method of simultaneous measurement is used as a
tool for fast determination of hydraulic and pneumatic properties of porous media.
With this new setting, the influence of soil air on water transport, especially the
range of saturation where the Richards approximation could be applied, should be
investigated. Pneumatic conductivity of the sample can be determined to get an
insight in gas transport properties. For this study, the simultaneous measurements
are applied to laboratory size columns of different types.
A major experimental hurdle is to keep the pore space stable for a prolonged
time. This is difficult for high water saturations because clay minerals can cause
soil swelling and the typically high water fluxes cause internal erosion. In addition,
there is always the problem of microbiological growth and associated clogging of soil
2
pores. Since these processes cannot be suppressed completely in soils, the problems
are circumvented altogether by working with columns made of sintered glass beads
which are rigid by construction and which can be treated with chemicals to both
prevent the growth of microorganisms and to remove possible precipitates.
The influence of the structure of a sample on multiphase processes is examined by
a set of different homogeneous and heterogeneous repacked sand columns. Hysteresis
of the hydraulic and pneumatic relations can be investigated at different scales with
a focus on textural and structural influence on the total effect.
The measured air conductivity data is simulated with three types of models which
are based on different fundamental assumptions. A comparison of the simulation
results show the advantages of the different approaches and the limitations of each
air conductivity model.
Experimental Background for Multiphase Phenomena
Investigations of multiphase phenomena have been subjected to the inquiry of petroleum engineers in the first half of the 20th century when the parallel flow of oil,
gas, and water in oil reservoirs was investigated (Muskat and Meres 1936, Leverett
1939). Their model was based on the assumption of local equilibrium, according to
which the relative phase conductivities and the capillary pressure could be expressed
through the universal functions of local saturation.
In the soil science, the Richards approximation could conveniently describe fluid
transport with a relatively simple model, therefore much investigation was carried
out on the soil water phase only. With the simple parameterizations of the functional
relationships of hydraulic properties it has been possible to simulate water-flow
in unsaturated porous media and come to a good agreement with measured data
(Brooks and Corey 1966, van Genuchten 1980, Burdine 1953, Mualem 1976).
The influence of the soil air phase has been subjected to various hysteresis theories
in which it was not possible to neglect soil air. The change of hydraulic conductivity with the history of the examined samples was reported by Kozeny (1927), who
explained the hysteresis effect by air entrapment. The development of entrapped
air was described by Miller and Miller (1956) for the different behavior of air-water
interfaces during drainage and imbibition of a porous medium. Several studies were
carried out for the investigation of the influence of hysteresis on the moisture retention characteristic (e.g. Topp 1969, Parlange 1976). The models developed for this
phenomena are reviewed in Viaene et al. (1994). The influence of entrapped air on
the conductivity relations were also investigated by different authors. The hysteresis
of the pressure-conductivity relation was obvious as it is directly connected to the
pressure-saturation relation. Hysteresis of the saturation-conductivity relation has
been observed with two different results. The experimental data of Poulovassilis
(1969) and Topp and Miller (1966) found a hysteretic behavior, whereas Nielsen
et al. (1986) and Kool and Parker (1987) found negligible influence of hysteresis on
the relation.
3
1 Introduction
A mathematical description of hysteretic water flow was developed by Parker et al.
(1987) and Lenhard and Parker (1987) for both the hysteretic retention characteristic
and the saturation-conductivity relation. Their model was based on a scaling of the
distinct phase saturations, where the entrapped part of a phase was separated from
the mobile part. Experimental observation of air entrapment effects on hydraulic
functions confirmed this approach (Stonestrom and Rubin 1989a) and demanded an
explicit consideration of the soil air phase in experiments (Faybishenko 1995, Fischer
et al. 1997, Dury et al. 1999).
Since transient methods for soil analysis became popular during the last decade
the so-called dynamic effects have been revived. The effect was reported by Topp
et al. (1967) who found differences in pressure-saturation relationship for static,
steady state flow and transient flow measurements. At a given pressure water contents were significantly higher for the static, or steady state conditions than for the
transient. A review of the different studies in this field can be found at Hassanizadeh
et al. (2002). The mechanisms which lead to dynamic effects in water flow can be
summarized:
ˆ Dynamic effects have been investigated experimentally at different scales. At
the pore scale Lu et al. (1994) have found for the capillary rise of water that
the fluid exhibits a jump behavior, which crucially depends on the wetting
state of the pore surface.
ˆ Water entrapment during drainage: water in smaller pores is isolated, which
results in larger saturation. Relative conductivity is expected to be lower, as
the mobile portion of water is decreased. This is in contrast to observations
(Wildenschild et al. 2001).
ˆ Pore water blockage near the bottom boundary during drainage. At large
pressure steps the lower layer of the sample is drained and, assuming air availability, a capillary barrier is built (Gee et al. 2002). Relative conductivity is
expected to be lower.
ˆ Wildenschild and Hopmans (1999) investigated the rate dependence of unsaturated hydraulic properties, performing one-step and multistep outflow experiments. They found that water content was higher for fast drainage with large
pressure steps. Dynamic effects were less significant in fine-textured media.
ˆ Air entrapment for drainage at high saturation. The water can only be drained
if air can replace it. For non-continuous air phase, water cannot be drained
effectively which results in higher saturation under dynamic conditions.
ˆ Microscale air phase displacement was investigated by Mortensen et al. (2001)
in a two-dimensional sample, where the non-uniqueness of flow properties was
a result of fast air finger flow and slower air back-filling.
4
ˆ Friedman (1999) described the differences between static and transient measurements by the dynamic contact angle instead of a static one. For a given
capillary pressure, the degree of liquid saturation decreased with increasing
liquid flow velocity, for wetting processes and vice versa for drainage.
With this study, hydraulic and pneumatic properties of porous media will be
examined simultaneously within one single transient experiment in order to have a
full set of material properties. The complex procedures of determination of either
the hydraulic or the pneumatic properties will be simplified to get an instrument
which can be routinely used.
5
2 Theory of Immiscible Fluids in
Porous Media
To compile the theoretical and experimental framework for water- and gas-transport
in unsaturated porous media, there is a need for some definitions of porous media, the
scales of interest, and the conceptual models which are used to describe multiphase
processes on the respective scales (Sections 2.1 - 2.3). A specific effect of multiphase
processes is the history dependence of the system. The hysteresis effects on hydraulic
functions is examined more closely in Sections 2.2.4 and 2.3.5.
The parameterizations which describe the soil hydraulic and pneumatic properties
are introduced in Section 2.4 followed by the mathematical description of multiphase
flow processes in Section 2.5.
2.1 Porous Media
Porous media occur in various forms, e.g. as soil, rock, sand, paper, porcelain, or
any kind of filter. These materials are characterized by their microscopic structure
which exhibit a large number of void spaces. Voids which are interconnected, are
called pores and a network of pores enable fluids to occupy them and move through
the pore system. Multiphase flow processes in the case of this study describe the flow
of the two immiscible fluids water and air, which can be observed at different length
scales, where flow processes are governed by specific fundamentals. Transitions
between scales are possible if a material property, which can be observed at a small
scale, can be averaged to get an effective property at the larger scale, e.g. the surface
tension at the micro scale has an impact on the capillarity at the pore scale (see
equation 2.5). Figure 2.1 gives an overview of the typical length scales (molecular,
pore, and continuum scale) which are common in literature. With a focus on fluid
flow, each scale has a specific structure, which are the elements comparable in size
with the scale of observation, and the texture, which describes the influence of
elements much smaller than the observed scale (Vogel and Roth 2002).
6
PSfrag replacements
transition
2.1 Porous Media
air phase
σwa
water phase
solid phase
tinuum scale
pore scale
L
L1
L1
L8
L8
L
averaging
olecular scale
L9
L7
L6
Figure 2.1: Typical length scales in porous media (blue: water; red: air; black: solid)
Molecular Scale
Molecular scale in this context means a length scale of Ångstrøm (10−10 m). At
this scale, the focus is on molecular interactions between the fluid particles and
the solid material (Figure 2.1 left). Forces between the particles are primary based
on electrical interactions. The interactions lead to effective properties like wetting
behavior of the matrix, surface films, and sorption of water and solutes. Adsorption
of water in films at the solid surface is influenced by surface properties. Besides
the influence of hydrophobic or hydrophilic surfactants, the geometry of the surface
has an impact on the evaporation of water as the radii of the menisci reduce the
steam pressure of water above the water air interface. For smooth surfaces, the
connectivity of water can be lost during the drainage process and ”pendular rings”
which are isolated portions of water, will develop. This effect is expected to be much
less pronounced in the case of a rough surface which offers edges and wedges for the
water to leak out (Dullien et al. 1988).
Pore Scale
Pore scale is within the scope of micrometers (10−6 -10−3 m). The individual molecules
are averaged to an ensemble of particles, which is described in a statistical way (Figure 2.1, center). The molecular scale state variables (e.g., velocity, molecular mass,
etc.) are averaged to pore scale ensemble properties like density, pressure, or temperature.
With the known geometry of the solid phase and the accompanying surface properties, the motion of the fluid phases can be described by the Navier-Stokes equation.
This depends on the mean free path length being much smaller than the diameter of
the pores. Numerical solution methods for this approach are limited by the speed of
computers available to small sample volumes because of the high spatial resolution
required.
Alternatively, processes can be described with pore network model simulations
using simplified models and geometry, e.g. the Young-Laplace equation applied to a
7
2 Theory of Immiscible Fluids in Porous Media
capillary network. These are a kind of shortcut in determining effective parameters,
as they reduce the porous medium and the fluids to a few effective properties (a review of several pore scale modeling approaches can be found in Blunt (2001)).
Continuum Scale
Figure 2.1 (right) shows an example for the continuum scale which is represented by
a column scale sample. Phases are no longer assigned to special locations. They are
averaged and distributed over the whole space with their specific fractions. Influence
of the pore geometry and surface properties and interactions between fluids are
included in the relevant effective state variables. Effective properties are influenced
by the texture and the structure of the sample. The example shows a column
with three different constituents which have their respective material properties.
Sampling of these materials result in new effective material properties of the entire
column.
2.1.1 REV, Averaging, and Continuum Approach
Extensive variables (e.g. bulk density or phase contents of a soil sample) have to
be averaged on a specific sample volume. If the mean value becomes constant with
increasing volume of observation, i.e. the influence of microscopic structure elements
can be described by a macroscopic property, then the value where it has become
stable is called the ”representative elementary volume” (REV) (Bear 1972). This
concept is necessary for the examination of larger soil volumes where the specific
phase geometries become too complex for explicit observation. Small scale variables
L3
L4
PSfrag replacements
croscopic parameter
volume
no defined
REV
REV1
REV2
scale
transition
L1
L6
L7
L8
L5
L2
Figure 2.2: Definition of a representative elementary volume. The threshold when
the observed value is constant with increasing volume is called an REV
with respect to the measured property.
8
2.2 Capillarity
are averaged to get effective material property at the scale of interest. Different
variables can have different REVs in the same porous medium. For some parameters
there are several plateaus in the parameter-value vs. observed-volume curve. This
means that they have different REVs at different scales. Figure 2.2 shows a soil
with a discrete hierarchical structure resulting in REV1 and REV2 . Knowledge of
an REV at the observed volume is necessary in making transitions between different
scales possible. For the example in Figure 2.1 (right) the first REV could be assigned
to the texture of the single materials and the second to the entire column.
The upper curve in Figure 2.2 shows another example. For this parameter there
is no REV within the range of the observed volumes. Averaging methods used for
transitions between scales cannot be applied here and maybe a REV can be found
by further increasing the sample volume.
2.1.2 Porosity, Liquid Content and Saturation
In an unsaturated porous medium, two or more fluids share the same pore space.
Each fluid occupies its respective partial pore volume fraction. The most common
approach is to distinguish between wetting and non-wetting fluids. For this study
the fluids of interest are water, the wetting fluid which is indexed as w, and air, the
non-wetting fluid denoted as a. In this two-phase system, the volume fraction of the
total pore volume (defined as the porosity φ of the medium) occupied by one of the
liquids is called saturation S.
The volumetric water content is defined as:
θw = Sw φ [−]
(2.1)
The volumetric air content is:
θa = Sa φ [−]
with:
X
(2.2)
θi ≡ φ where i = a ; w
(2.3)
Si ≡ 1 where i = a ; w
(2.4)
i
and
X
i
2.2 Capillarity
As the macroscopic distribution of water and air in porous media depends on microscopic effects, a zoom into the smaller scales with their governing processes is
necessary.
9
2 Theory of Immiscible Fluids in Porous Media
2.2.1 Micro Scale Effects
Surface Tension
Forces between molecules of a single phase are cohesive forces. For molecules at the
interface between two immiscible fluids, the forces are stronger within its own fluid
than to the other fluid. The effect is sketched in Figure 2.1 (left) where the forces on
several fluid particles are indicated by arrows. The forces on the particle in the center
is balanced whereas the particle at the left water-air interface is drawn back into the
water phase. This imbalance of forces gives the surface its specific curvature, since
the system as a whole has the tendency to reach a condition of minimum energy.
The enlargement of the surface A demands some energy EA which is stored in the
surface and is given back when the surface area is reduced. The energy per area is
defined as specific surface tension σ (Gerthsen et al. 1992):
Ea = σ · A
2.2.2 Capillarity Effects at the Pore Scale
Wettability
Different fluids (air, water, oil, etc.) show different adsorption behavior on the soil
matrix. The microscopic interactions are summed up to the macroscopic surface
property which is called wettability.
Figure 2.3 shows two immiscible fluids in contact with a solid surface. The angle
between the fluid-fluid interface and the solid surface is called the contact angle
which is defined for steady state conditions as:
γ=
σas − σws
σwa
(2.5)
PSfrag replacements
non-wetting
wetting
solid
σws
σas
σwa
γ
L1
L6
L7
L5
L4
L2
L3
Figure 2.3: Definition of surface tension and contact angle in a three phase system
10
2.2 Capillarity
L1
L2
L4
PSfrag replacements
σwar
γ
pa
pw
L3
L5
Figure 2.4: Surface element in a capillary
The indices s, a, and w, represent the solid, air, and wetting phase, respectively.
For example σwa is the surface tension of water with respect to air. Equation 2.5 is
sw
known as Young’s equation. If σsnσ−σ
> 1, it is not possible to have a steady state,
nw
as the wetting fluid spreads over the solid surface. For a contact angle γ (Figure
2.3) less than 90 ◦ the fluid is called wetting w, for larger contact angles it is called
non-wetting phase, a (air) (Bear 1972).
Another aspect related to wettability is the speed at which fluids spread over a
solid surface. This speed is affected by the surface tensions and viscosity of the fluids.
Spreading speed is increased by lower surface tensions or by lower viscosities.
Capillary Pressure
At the pore scale the porous medium can be idealized as an accumulation of capillary
tubes. If two immiscible fluids are in contact with each other and equilibrium is
reached, the pressure of the non-wetting phase pa at the interface between the fluids
is greater than the pressure of the wetting phase pw (see Figure 2.4). The difference
between the phase pressures of the two fluids is called capillary pressure:
pc = pa − pw
(2.6)
11
2 Theory of Immiscible Fluids in Porous Media
The Young-Laplace equation for capillarity defines the curvature dependent capillary
pressure as:
pc =
with
2σwa cos γ
r
(2.7)
σwa surface tension of water with respect to air
γ
contact angle
r
radius of the observed meniscus
Capillary pressure depends on the one hand on the geometric properties of the pore
(radius r) and on the other hand on the surface tension (contact angle γ) which is
a function of the chemical compositions of the fluids and the solid surfaces.
All the above assumptions were made under the precondition that there is only
pure water in contact with simple solid surfaces. The influence of solutes in water
or surfactants at solid surfaces is not incorporated in this view.
Potentials
Potential energy of soil water, the so-called soil water potential, is a summary of
several partial potentials:
The gravitational potential is the energy density of water at a position z moved
from a reference state z0 :
ψg = −ρw g(z − z0 ) for ρw = const.
(2.8)
The tensiometer potential is the sum of hydrostatic potential, pneumatic potential,
and matrix potential.
ψtp = ψh + ψp + ψm
(2.9)
Pneumatic potential is only necessary if air pressure in the porous medium is not at
atmospheric pressure, i.e. air bubbles are entrapped and do not communicate with
an air reservoir.
The osmotic potential is the energy density required for a change of solute concentration in a fluid. As mentioned above, only pure water has been used for this
study which makes osmotic potential obsolete for further consideration.
2.2.3 Capillarity Effects at the Continuum Scale
In a multiphase system there exists a fundamental connection between saturations
of the wetting phase and the non-wetting phase, and capillary pressure. Figure 2.5
shows a pore at two different capillary pressures pc1 (solid) and pc2 (dashed). For the
pressure pc1 the interfacial area has the curvature radius r1 and the wetting phase
12
2.2 Capillarity
L5
PSfrag replacements
water
solid
air
interface at pc1
interface at pc2
L1
L2
L4
L3
Figure 2.5: Model of pore cross-section with two different states of saturation of
water, corresponding to the pressures pc1 and pc2 and their respective
interfaces (Corey 1994)
saturation Sw1 . With decreasing capillary pressure (pc1 → pc2 ), the wetting phase is
drawn back to that part of the pore with smaller dimensions and the new interface
radius r2 corresponds to smaller saturation Sw2 . Therefore, saturation is a function
of capillary pressure. The pressure which is necessary to start drainage of a waterfilled pore is called air entry pressure pe . Since interconnected pores have different
diameters, the desaturation of the pore space occurs in discrete jumps depending on
the largest pore available for drainage (Corey and Brooks 1975).
Macroscale capillarity in analogy to the microscale capillarity is a function of
surface tension σ, contact angle γ, and via pore radii r, of the pore geometry. This
results in a relation between capillary pressure and saturation:
pc = pc (Sw )
(2.10)
which is the so-called soil water characteristic.
2.2.4 Hysteresis in Soil Water Retention Function
The soil water characteristic is not a unique static function since the pc (Sw ) relationship depends on the pressure (or saturation) history. If the response of the porous
system depends on the history of the driving forces which imply these responses,
this kind of dependency is called hysteretic behavior. This means, in the case of
the capillary pressure relation, that the drainage behavior (non-wetting phase is
repelling the wetting phase) differs from the imbibition behavior (wetting phase is
repelling non-wetting phase). To predict transport properties, knowledge of this
pattern is necessary. The distribution of phase saturation has a strong impact on
the flow paths available for solute displacement.
13
2 Theory of Immiscible Fluids in Porous Media
L1
L2
PSfrag replacements
γimb.
γdrain.
γimb.
> γdrain.
L3
Figure 2.6: Raindrop effect visualizing dependence between contact angle and wetting front behavior. The drop is running down a homogeneous surface.
The zoom on the right side shows the difference between the advancing
(bottom) and receding wetting front (top).
The physical basics of hysteresis have to be viewed on different scales and their
macroscopic results are the total of different effects.
Micro Scale Hysteresis
On the micro scale, the contact angle was the parameter including all surface fluid
interactions (Figure 2.3). For the dynamic case, the correlation between contact
angle and the mode of displacement is shown in Figure 2.6. The drop is running
down a solid surface. At the lower end one can see the large contact angle of the
advancing front, where the dry solid has to be wetted. At the upper end the receding contact angle is much smaller as the solid was originally wet. The effect is
known as ”raindrop effect”. The difference between the static contact angel (Figure
2.3) and the dynamic one (Figure 2.6) strongly depends on the displacement velocity. For further details about the influence of dynamic contact angle see Friedman
(1999).
Pore Scale Hysteresis
Pore scale hysteresis is the result of two major effects. The first, is the so-called
”ink bottle effect”, which is sketched in Figure 2.7. The two capillaries (a) and (b)
each have their specific height of capillary rise. For capillaries with non-uniform
diameter, saturation history has an impact on the height of capillary rise. Figure
2.7 (c) shows drainage of the irregular capillary. The receding water table is blocked
at a pore neck. (d) shows the imbibition case, where the advancing water table is
blocked by the pore body and the height of capillary rise corresponds to that of the
larger tube (b). Although the same pressure is applied to all capillaries, the
14
2.2 Capillarity
a
b
c
d
PSfrag replacements
a
b
c
d
e
trapped
air
Figure 2.7: Model of drainage and imbibition heights of capillary rise. (a) Capillary
rise in small tube equivalent to pore necks. (b) Capillary rise in large
tube equivalent to pore bodies. For mixed pore diameters, capillary rise
in drainage (c) differs from imbibition (d) (Dullien 1979).
drained tube (c) contains obviously more water than the imbibed (d).
The second effect, causing pressure saturation hysteresis, is phase entrapment.
Displacement of one fluid by the other, can break the continuity of the displaced
fluid by separating portions of it, which are then enclosed in the invading fluid.
Two mechanisms which end in phase entrapment are sketched in Figure 2.8 and
2.9.
The ”snap-off” effect describes an inclusion of non-wetting phase bubbles in the
pore bodies. It depends on the surface properties (wettability) and on the ratio
between the sizes of pore bodies and pore throats.
The effect of ”by-passing” is shown in Figure 2.9 where an initially dry pore system
is imbibed. The smaller pore is filled with water due to its capillarity, while air in
the larger pore is separated from the air continuum.
Entrapment is not limited to the gaseous phase, as water cut off from the communicating liquid phase is also entrapped. This water is placed in small pores which
15
2 Theory of Immiscible Fluids in Porous Media
are surrounded by larger ones, or in ”pendular rings” which are shown in Figure 2.5
as the pc2 state.
L1
PSfrag replacements
e pore body to neck ratio
l pore body to neck ratio
water
air
pore
throat
pore
body
L2
L3
L5
L6
L7
L8
L4
Figure 2.8: Influence of the pore geometry on the inclusion of a fluid caused by the
”snap-off” effect. (Chatzis and Dullien 1983)
L1
L2
PSfrag replacements
before inclusion
after inclusion
Figure 2.9: Influence of pore geometry on the inclusion of a fluid caused by the
”by-passing” effect. (Chatzis and Dullien 1983)
Macroscale Hysteresis
Figure 2.10 shows the pressure saturation relation for a macroscopic soil sample
(φ = 0.32). The special cases of the hysteretic retention curves are classified in
three categories: primary, main, and scanning curves with their respective drainage
and wetting branches:
16
2.2 Capillarity
3.0
2.5
PSfrag replacements
φ = θw + θa
θw,r
θa,r
φ − θw,r − θa,r
Pressure Head [m]
PDC
PWC
MDC
MWC
SDC
SWC
L5
2.0
L7
L6
L9
1.5
L8
L10
L11
1.0
0.5
0
0
0.20
0.10
0.30
L1
L2
L4
L3
Figure 2.10: Hysteresis in the pc (S) relationship. Solid lines refer to drainage processes and dashed lines to imbibition. Primary curves (red) start with
full saturation of the respective fluids, whereas main curves (blue) fix
the range of measurement starting with unsaturated conditions. The
scanning curves (green) can start at any point within the limits of PDC
and PWC when the sign of ∂h/∂t changes.
ˆ Primary drainage curve (PDC): The sample is fully saturated with water and
the applied capillary pressure is pc = 0. Decreasing capillary pressure starts
drainage when pressure reaches the air entry value, pe , which is the pressure
corresponding to the largest pore connected to the air reservoir. Minimum
saturation, the residual water saturation θw,r is reached, if any further decrease
in capillary pressure results in a negligible decrease in saturation. It is a
result of an unconnected water phase and the water remaining in the sample
is associated with surface films or pendular rings. Since phase continuity is
lost, it is not possible to remove this amount of water by a further decrease in
capillary pressure.
Primary wetting curve (PWC): The totally dry sample (complete air satura-
17
2 Theory of Immiscible Fluids in Porous Media
tion) is saturated with water by increasing capillary pressure. Due to phase
entrapment effects, the maximum water content θw,s is less than porosity and
the difference θa,r = φ−θw,s is the maximum residual air content which cannot
be removed by increasing capillary pressure as phase continuity is lost.
ˆ Main drainage and wetting curve (MDC, MWC): The main curves form the
hydraulic limits of the fluid distribution. It is not possible to obtain fluid
contents outside these limits by normal hydraulic fluid flow, i.e. fluid flow
governed by Darcy’s law. MDC and MWC approach each other asymptotically
when h → 0 and h → ∞.
ˆ Scanning drainage and wetting curve (SDC, SWC): Scanning curves are the
typically measured retention curves of undisturbed soils in laboratory. They
start at any point within the range derived by the main curves1 and also end
at any point.
The different ranges of water content and the resulting effective saturations are
described by (Luckner et al. 1989):
Sw =
θw − A
φ−A−B
(2.11)
with the scaling factors A and B which are summarized in Table 2.1 representing
primary and main curves.
A
B
PDC PWC MDC
θw,r
0
θw,r
0
θa,r
θa,r
MWC
θw,r
θa,r
Table 2.1: Values of the scaling factors A and B for PDC, PWC, MDC, and MWC.
Hysteresis influenced by macroscale structure elements
The effect of pressure saturation hysteresis is also caused by macroscale structures at
the continuum scale. In samples with heterogeneous structure, a type of large scale
”snap-off” can be observed. For example, a fine-textured lens should be embedded
in a coarse-textured material. Drainage of this sample should be induced, by a fast
decrease of capillary pressure to a value less than air entry pressure of the coarse
material and higher than the air entry pressure of the fine material. Drainage of the
coarse material could disconnect the fine lense from the water phase which would
cause drainage by further decrease of capillary pressure to be impossible, or it could
1
Investigations carried out with undisturbed soil samples usually do not start at full saturation
of one phase and therefore measurement do not result in a primary curve.
18
2.2 Capillarity
drastically reduce hydraulic conductivity which would cause drainage to be very
slow. For coarse-textured lenses embedded in fine-textured material the opposite
effect is observed. During imbibition of an initially dry sample pneumatic contact
from the lenses to the air reservoir is lost and air is entrapped within the lenses. Both
effects occur even more in the fast transient multistep outflow experiments where
diffusion of the involved fluids and phase transitions have only little impact.
2.2.5 Experimental Methods to Determine Pressure-Saturation
Relation
There are several standard methods for determination of the pressure saturation
relation. They can be sorted according to increasing precision:
ˆ Long column: A long homogeneous column with a water reservoir at the lower
end is allowed to reach equilibrium. Forces acting on the fluids are result of
gravity and capillarity. Saturation is determined at several heights by slicing
the sample. The discrete resolution of saturations is limited to a pressure
range, limited by the height of the sample.
ˆ Centrifuge: The small initially saturated sample is placed in a centrifuge,
where it is turned around with a specific velocity. After some time the centrifugal and capillary forces reach equilibrium and the saturation can be determined. This method is good for determining residual wetting fluid saturation.
The pressure range depends on the power of the centrifuge and the mechanical
stability of the sample.
ˆ Hanging water column: The sample is placed on a water saturated porous plate
with a high air entry pressure. Pressure is applied via water phase by a water
column of specific height. After reaching equilibrium saturation is measured
gravimetrically. With this method a maximum water phase pressure of about
300 cmWC 2 can be applied.
ˆ Pressure cell: The sample is placed on a porous plate inside a pressure container. A high air pressure is applied to the container, which induces displacement of water. The system is allowed to reach equilibrium and saturation
is measured gravimetrically. This method can be very time consuming as the
equilibrium state is reached after many days (depending on actual saturation).
With decreasing of the applied pressure (up to some ten bars), the state of
water is not clear anymore which brings some difficulties in the interpretation
of the results (Klute and Dirksen 1986).
2
The water potential is indicated in cmWC throughout this study, which is the equivalent of a
hanging water column.
19
2 Theory of Immiscible Fluids in Porous Media
ˆ The Brooks’ method (Brooks 1980) works the opposite way. Saturation is
controlled by adding a defined portion of water to the sample. After reaching
equilibrium the achieved capillary pressure is measured.
2.2.6 Empirical Descriptions for Pressure-Saturation
Relation
The empirical approaches try to describe the measured data with a simple mathematical formulation with the aim of interpolating the retention curve between the
measured data points. There are several models developed by Brooks and Corey
(1964), Brutsaert (1967), Campbell (1974), van Genuchten (1980), Lenhard et al.
(1993), etc.. In contrast to the purely empirical models, which could only be applied
to a specific soil, these models fit experimental data by using material parameters
which reflect the geometric structure of the pore space. Parameters are related to
pore size distribution, air entry pressure, etc.. Two of the most commonly used models should be introduced here with special attention to their application of hysteretic
problems.
2.2.6.1 The Brooks and Corey Model
According to Brooks and Corey (1964), saturation of a wetting fluid in porous media
on a drainage cycle typically decreases with capillary pressure as indicated by the
relationship:
µ ¶λ
pe
θw − θw,r
Se ≡
=
for pc ≥ pe
(2.12)
θw,s − θw,r
pc
where Se
θw
θw,s
θw,r
λ
pc
pe
effective water saturation
volumetric water content
maximum water content
residual water content
pore size distribution index
capillary pressure
air entry pressure
or conversely:
1
−λ
pc = pe · Se
for pc ≥ pe ,
θw ≥ θw,r
(2.13)
The parameter λ is related to the pore size distribution of the sample. Brooks
and Corey found that for a typical porous medium, λ is about 2. Soils with welldeveloped structure have values of λ less than two and sands normally have values
20
2.2 Capillarity
of λ greater than 2, sometimes larger than 5. A large λ means narrow pore size
distribution; small λ is an indicator for a very heterogeneous pore structure.
The physical meaning of the parameters Sw,r , and pe is normally overrated. For
a homogeneous isotropic porous material, the ”residual water content” is defined as
the water content which provides the best fit to a straight line for water contents
greater than the critical air entry pressure pe , in a lnSw,e vs. lnpc plot. Actually,
they should be seen as fitting parameters. Corey and Brooks (1999) explained the
appropriate applications and limitations of their model as they found it to often be
misused.
2.2.6.2 The van Genuchten Model
The van Genuchten (1980) model describes the pressure saturation relation in terms
of the matric head
h=−
2σwa
ρw gr
(2.14)
which is the relation between capillary radius r and the rise h of the water, as:
Se ≡
θw − θw,r
= (1 + (αh)n )−m
θw,s − θw,r
where Se
θw
θw,s
θw,r
h
n, α, m
(2.15)
effective water saturation
actual water content
maximum water content
residual water content
matric head: h = − ρψwmg
fitting parameters
For a closed form of descriptions of hydraulic functions (see Section 2.4.1) it was
convenient to couple the parameters m and n by the condition3
m = 1 − 1/n
which leads to
−1+1/n
Sw (h) = [1 + [αh]n ]
(2.16)
or inverted:
pc (Sw ) =
3
¢1
1 ¡ −n/[n−1]
−1 n
Sw
α
for pc > 0
(2.17)
This approximation has no physical significance and was only done for mathematical convenience
since integration of the hydraulic conductivity function became less complicated.
21
2 Theory of Immiscible Fluids in Porous Media
a
PSfrag replacements
logSw [-]
log (ψ [ cmWC ])
a
b
0
0.9
-0.5
-1.0
0.8
-1.5
lth
n Genuchten (1980)
s and Corey (1964)
0.7
theta
Sw [-]
g (ψ [ cmWC ])
b
1.0
-2.0
-2.5
0.6
-3.0
0.5
-3.5
0.4
-4.0
2
0
0.3
G
4
6
lpsi
0.2
B
0.1
0
0
1
2
4
3
5
6
psi
Figure 2.11: Comparison of the two parameterizations of the retention function. The
Brooks and Corey relation (red) has a well defined air entry pressure
whereas the van Genuchten relation (blue) has a smooth decrease of
saturation with increasing potential. (b) is a log-log plot of (a).
where n has to be larger than 1.
In contrast to the Brooks and Corey model, all capillary pressures greater than
zero are taken into account even if they are less than pe . The parameter α corresponds to the displacement pressure while the pore geometry is represented by
parameters n and m, which describe the slope of the curve. For the closed formulation, the parameter m is derived from n.
A correspondence between the Brooks and Corey and the van Genuchten models was deduced by Lenhard et al. (1989) (Figure 2.11). They compared the two
parameterizations by using the differential fluid saturation capacities, ∂Se /∂pc for
pc > pd or pc > 0, respectively:
λ+1
λ
∂Se
=
· Se λ
∂pc
pd
(2.18)
¢
¡
∂Se
= α · m · n · Se−1/m − 1 · Se1/m+1
∂pc
(2.19)
and
The two variables pd and α are eliminated by insertion of equations (2.13) and (2.17)
in (2.18) and (2.19) respectively, which results in:
∂Se
λSe
=
,
∂pc
pc
22
(2.20)
2.2 Capillarity
¡
¢
∂Se
m
=
· Se−1/m − 1 · Se1/m+1
∂pc
(1 − m)pc
(2.21)
Both models correspond to each other for Se = 0.5, which results in:
λ=
¢
m ¡
1 − 0.5 1/m
1−m
(2.22)
Correspondence of entry pressure pe and α is achieved by:
pe =
1/λ
¢1−m
Semp ¡ −1/m
Semp − 1
α
(2.23)
with the empirical approach for the saturation of maximum correspondence:
Semp = 0.72 − 0.35e−n
4
(2.24)
Comparison of the two parameterizations in Figure 2.11, shows the conceptual
differences of the models. The Brook and Corey relationship could be deduced if
one assumes that the geometry of the pore space occupied by the wetting fluid is
fractal, i.e. differing only in scale, at all water contents greater than θw,r (Miller
and Miller 1956). Additionally the models assume a distinct air entry pressure.
This becomes clear in a log-log plot as show in Figure 2.11 (b) where the retention function is represented by a straight line over a specific range beginning at
the entry value. Although this model was based on physical assumptions, it had
disadvantages in numerical simulations because the discontinuity caused by the entry pressure led to difficulties in the simulation process because of the non-steady
derivative at the entry pressure; additionally the function is costly to evaluate. The
van Genuchten model deduced the smooth function with the assumption that the
pore size distribution started with pores of infinite radius. The idea behind that
was the slow development of the capillary fringe, where water, drained from the
large openings of the pores resulted in an outflow before the ”air entry” pressure
had been reached. Although the Brooks and Corey model was better related to
physical principles, the van Genuchten model has been established because of its
better numerical applicability. The derivative dθ/dh of this function is continuous
and becomes asymptotically zero toward the fine and large pores. A lot of research
was done based on the van Genuchten parameterization and almost all refinements
in the theoretical field, like non-wetting phase models, hysteresis, etc., were tested
with this model.
2.2.6.3 Pressure-Saturation Hysteresis Models
Since the water retention characteristic is of fundamental importance for the soil
water behavior, the effect of hysteresis was examined by several authors with different concepts. Adequate parameterizations for the hysteretic pressure-saturation
relation shown in Figure 2.10 were subject to different approaches with the aim of
23
2 Theory of Immiscible Fluids in Porous Media
predicting any scanning curve with the knowledge of the main curves. The conceptual models for hysteresis are based on the dependent domain theory which was
developed in the 1940’s and refined by Parlange (1976) and Mualem (1974). The
empirical models were invented by Scott et al. (1983) and further developed by Kool
and Parker (1987) and Parker and Lenhard (1987).
Dependent Domain Theory
The dependent domain theory is based on the assumption that the porous medium
consists of pores with variable pore size. Behavior of every pore during drainage
or imbibition depends on the state of the surrounding pores, i.e. a pore can only
be drained if air is available from a neighboring pore. The basic concept of this
model and its implications for the different behavior of the pore domains during
drainage and imbibition are shown in Figure 2.12. The model distinguishes between
two parameters that characterize the system: r1 , the radii of the openings of the
pores in a group, and r2 the radii of the pores within the group. If the capillary head
changes from h(r) to h(r+dr) during imbibition all pores having radii r ≤ r2 ≤ r+dr
are filled. In the drainage branch of the cycle, the head h(r) is reduced to h(r − dr)
and the pores in the groups with radii r1 in the range r ≤ r1 ≤ 1 having radii
r ≤ r2 ≤ r − dr are drained.
For predictions of scanning curves Mualem (1984) developed a scaling approach
which is calibrated by use of both the main drainage and the main imbibition curve.
The influence of the surrounding pores is accounted for by the macroscopic dependent domain factor. This factor describes the relative portion of drainable pores in
the actual branch dependent of the saturations at the point where the wetting direction had changed. For predictions of the scanning cycles, the complete history of the
system is necessary in calculating saturations (see Figure 2.12, right). Prediction of
rag replacements
a
b
c
0
1
r1
r2
drainage
wetting
a
0
r2
b
0
r2
w
c
0
r2
d
1
1
1
r1
0
1
r1
1
0
1
r1
0
Figure 2.12: Conceptual model of the dependent domain theory for the hysteretic
pressure saturation relation. Contour map of the filled pores for (a) the
primary wetting and (b) primary drainage branch, (c) wetting after six
processes of imbibition and drainage (Mualem 1974).
24
2.2 Capillarity
the imbibition curve is given by:
θi (h) = θ∆ + P (θ∆ ) [θw,s − θw (h∆ )] [θw (h) − θw (h∆ )]
(2.25)
and for the drainage curve:
θd (h) = θ∆ − P (θ) [θw,s − θw (h)] [θw (h∆ ) − θw (h)]
(2.26)
where the subscript delta indicates a reversal point, and superscript w denotes the
main imbibition curve. The domain dependent function P (θ) is given by:
P (θ) =
θw,s − θ
[θw,s − θw (h+ )]2
(2.27)
where h+ is the pressure head at which
θd (h+ ) = θ
with (2.17) this gives
h+ =
¤1/n
1 £ d −1/m
(Sw,e )
−1
α
For the imbibition process, P is a function of the known water content at the reversal
point, while for the drainage branch, P is a function of the unknown water content
θ and therefore the curve must be obtained iteratively.
Kool and Parker Model
The second type of hysteresis models are based on an empirical description of the
connection between main and scanning curves. With the simple model of Scott et al.
(1983) the drainage scanning curves are predicted by rescaling the main drainage
curve to pass through the residual water content (θw,r ) and the last reversal point
d
from wetting to drying (Sw,e
(h∆ )), by replacing the saturated water content (θw,s )
by:
∗
θw,s
=
d
(h∆ )]
θ∆ − θw,r [1 − Sw,e
d
Sw,e (h∆ )
(2.28)
The imbibition scanning curves are predicted by forcing the curve through the last
w
(h∆ )) and the saturated water content
reversal point form drying to wetting (Sw,e
θw,s by the replacement of residual water content:
∗
=
θw,r
i
(h∆ )
θ∆ − θw,s Sw,e
i
1 − Sw,e (h∆ )
(2.29)
25
2 Theory of Immiscible Fluids in Porous Media
d
where Sw,e
(h∆ ) effective saturation on the main drainage
curve at the reversal point pressure head
i
Sw,e
(h∆ ) effective saturation on the main imbibition
curve at the reversal point pressure head
water content at the reversal point
θ∆
θw,s
saturated water content
θw,r
residual water content
h∆
matric head at the reversal point
Since this model did not reflect air entrapment during imbibition the scanning
cycles are not closed. Kool and Parker (1987) combined this approach with an
empirical relation for air phase entrapment where the saturation of a scanning curve
is derived by
d
θw,s
− θ∆
d −θ )
1 + R(θw,s
∆
(2.30)
1
1
− d
w
d
− θw,s θw,s − θw,r
(2.31)
d
θu = θw,s
−
where
R=
d
θw,s
d
w
where θw,s
≥ θu ≥ θw,s
. This empirical equation provides a useful approximation for air entrapment and has the advantage that no additional parameters are
introduced. Furthermore, the model was expressed with the van Genuchten parameterization. The whole hysteresis loop could be described by the parameter
vector (θw,s , θw,r , αd , nd ) for the main drainage curve (MDC) and (θw,s , θr , αi , ni ) for
d
i
the main imbibition curve (MIC) with the simplification: θw,r
= θw,r
= θr and
d
i
θw,s = θw,s = θw,s . With the scaling of θw,s and θw,r the drying scanning curves were
∗
obtained by using the parameter vector (θw,s
, θw,r , αd , nd ) and the imbibition scan∗
ning curves were obtained by using the parameter vector (θw,s , θw,r
, αi , ni ). With
this model, the parameters for drainage and imbibition could be estimated independently.
Kool and Parker (1987) compared their model with the dependent domain model.
Simulations of hysteretic retention curves for 8 soils were carried out with both
models. The results showed no unacceptable loss in accuracy as long as the soils did
not have a narrow pore size distribution. The major advantages of this model are the
small number of parameters and the low computational costs for simulations.
26
2.3 Fluid Flux
2.3 Fluid Flux
Forces on the fluids e.g. due to a pressure gradient cause a motion through the solid
matrix if the fluid phase is continuous along the field. Fluid flow can be described at
the pore scale by the Navier-Stokes equation and by Darcy’s law at the continuum
scale. For a saturated medium the only resistivity to the fluid flow is offered by
the solid matrix. The so-called saturated conductivity is influenced by the pore
geometry and the fluid mobility. Conductivity of an unsaturated porous medium is
additionally influenced by the presence of other fluids as they share the same pore
space.
Conductivity at the Pore Scale
Figure 2.1 (center) shows the distribution of the phases in a partially saturated
porous medium. The presence of two fluids, reduces the available flow paths for
each fluid, which influences their transport properties. If air saturation is increased,
water is gradually drawn back to smaller pores which is concomitant with increasing
tortuosity of the flow paths.
2.3.1 Conductivity at the Continuum Scale
The dependency between a driving pressure gradient and the volumetric water flux
in a saturated porous medium was first observed by Darcy (1856) who deduced the
empirical relation:
jw = −Kw,sat ∇p
with
jw
Kw,sat
∇p
volumetric water flux
saturated conductivity
pressure gradient
(2.32)
4
For the one-dimensional vertical case it can be simplified to:
jw,z = −Ksat,z
∂p
∂z
The saturated conductivity is dependant on the fluid and the solid material properties. There is also a fluid independent property, called permeability or intrinsic
permeability which is related only to the influence of the porous matrix.
4
In some textbooks K is called conductivity and ki permeability. In other nomenclature K is
permeability and ki the intrinsic permeability.
27
2 Theory of Immiscible Fluids in Porous Media
The unique relationship between permeability ki and saturated hydraulic conductivity Kw,sat , is described by:
Kw,sat = ki
ρw g
ηw
(2.33)
where the subscripts of w are properties related to water and Kw,sat has the dimensions of [ML−1 T−2 ]. Using the fluid properties ρl and ηl of a different liquid l the
conductivity of the fluid can be calculated by this relation5 .
Permeability ki [L2 ] of the soil can be calculated via Darcy’s law from measurement
of saturated water flow according to:
ki = −
where ki
jw
ρw
g
ηw
∂p
∂z
jw,z ρw g
∂p
ηw ∂z
(2.34)
intrinsic permeability
water flux 6
density of water
gravitational acceleration
dynamic viscosity of water
pressure gradient 6
For unsaturated conditions the pore space availability of the water phase changes
as its saturation is changed. A simple approach was the extension of Darcy’s law
by Buckingham (1907):
jw = −K(θ)
∂p
∂z
(2.35)
where the measured unsaturated conductivity K(θ) includes all influences that have
an effect on water flux.
2.3.2 Experimental Methods of Determining Unsaturated
Conductivity
The measurement of unsaturated conductivity over a wide range of saturations is
essential for the characterization of a porous medium. Solute transport, as well as
the extraction of water or oil crucially depend on this property. In laboratory sized
samples, the measurement can be done in several ways. Most of the methods use
water as an active fluid. Of the large variety of methods available two different
5
Since gaseous phases do not act as true fluids in porous media special gas flow effects has to be
taken into account. See Section 2.3.4
6
jw and ∂p
∂z can be determined by measurement of the falling head (Section 3.3)
28
2.3 Fluid Flux
approaches will be explained here because they are based on different principles.
These can be distinguished into two separate categories, the steady state methods
and the transient methods.
Steady State Methods
The steady state methods are based on the evaluation of Darcy’s law (equation 2.32).
Stationarity in this context means that the volumetric water flux or the applied
pressure gradient is kept constant during the experiment. For the measurement of
every single point of the K(θ)- or K(ψ)-relation, the specific state of the system
(θ, ψ) has to be adjusted and a steady state has to be reached.
For the constant head method, the sample is connected to predefined tensions
at two opposite ends by porous plates. Tensiometers placed at different heights in
the sample are observing the pressure gradient. At one end water is applied with
a constant pressure. If a steady state is reached, the constant flux at the other
end is measured and hydraulic conductivity can be evaluated by equation 2.35. For
evaluation, the conductivity of the porous plates has to be considered (Klute and
Dirksen 1986).
The constant flux method works in such a way that the flux is controlled instead
of the pressure head. At one end of the sample, a constant flux is applied by
infiltration or evaporation. Tensiometers in the sample observe the evolution of a
unit gradient. As a steady state is reached, unsaturated hydraulic conductivity can
be evaluated.
Transient Methods
Transient methods are based on the evaluation of temporal evolution of the flux
process in the examined sample. Observation of the hydraulic heads is made by
several tensiometers and the water content is monitored by time domain reflectrometry (TDR), gamma attenuation, or if an invasive method is chosen by gravimetric
measurements.
For the instantaneous profile method, the sample has to be instrumented with the
observation tools (tensiometers, TDR, . . . ) and the initial state has to be determined precisely. The sample is drained from one end and the temporal evolution of
tensions and saturation is monitored. Evaluation of the data is done by balancing
the temporal evolution of the water content in the sample.
The most promising tools for transient determination of unsaturated hydraulic
conductivity are the stimulated out-/in-flow methods. For the most simple case, the
flux at one end of the sample is negligible and at the other end a pressure gradient is
applied to the water phase. The pressure can be changed in a single reduction (onestep experiment) or in several steps (multistep outflow experiment). The major
disadvantage of these methods is the small pressure range of the measurement.
Especially for the dry range, additional measurements have to be made, e.g. by
evaporation experiments. The multistep outflow method has been used for this
study and will be explained in detail in Section 3.1.
29
2 Theory of Immiscible Fluids in Porous Media
2.3.3 Parameterization of Unsaturated Conductivity
Since it is not always easy to perform the necessary experiments to measure unsaturated conductivity it is often more convenient to calculate the K(θ) relationship
from other properties of the medium which are easier to measure. The calculations
can also be used to check the experimental results (Brutsaert 1967). The saturation – conductivity relation, K(θ), has been parameterized by several approaches.
An overview on the geometric effects which have an influence on the unsaturated
conductivity is given in Figure 2.13. The models developed on this basis can be
sorted into two categories, the purely empirical models, and the geometry based
models.
2.3.3.1 Purely Empirical Conductivity Models
For the simpler models, the pore space is characterized by a bundle of homogeneous
parallel capillaries, each with the same radius. There are two different views of this
model configuration. One idea is that the tubes are drained in their centers and
water is only at the capillaries walls. The model which describes this problem was
analyzed by Averjanov (1950). This model was based on the similarity of Darcy’s
law and the Hagen-Poiseuille equation:
Darcy: jw = −
ki
∇p ;
ηw
Hagen-Poisseuille: jw = −
r2
∇p
8ηw
(2.36)
A comparison of the two equation gives the conductivity of a single tube with radius
r which can be extended for a bundle of capillaries:
single tube: ki =
r2
8
;
bundle of capillaries: ki =
φ · r2
,
8
where φ is the porosity and r is the mean radius of the capillaries in the bundle. For
an annular flow of the water in the single tube the conductivity was approximated
by a power type relationship:
n
kw (Sw,e ) = ki Sw,e
(2.37)
where the saturation of the annular water at the boundary is given with respect to
the air-filled core of the tube (radius ra )
Sw,e = 1 −
³ r ´2
a
r
The other view assumes an effective hydraulic radius of the pores in a granular
medium, where pore cross-sections of any geometry can be included. This model
30
2.3 Fluid Flux
L5
L2
L2
PSfrag replacements
PSfrag replacements PSfrag replacements
PSfrag replacements
apillary bundle
le
l
r
tortuosity
L1
l
r
par
L4
L4
L3
tort
L1
r1l
r2
L1
l
rr1
r2
uniform radiicut and rejoined
uni
L1
L5
diff
Figure 2.13: Geometric effects influencing permeability
was developed by Kozeny (1927) who introduced the tortuosity parameter τ because
the irregularities of the pores had to be accounted for in the model. The tortuosity
parameter τ was used as a measure of the geometric shape information. Actually
tortuosity represents the influence of the winding of the fluid-filled pores through the
matrix and accounts for the change in the topology of the fluid phase as saturation
changes. In drainage processes where the water phase is drawn back to the smaller
pores, tortuosity increases since the high connectivity of these pores result in longer
microscopic path lengths (see Figure 2.1, center, and Figure 2.13). The resulting
conductivity function is of the power law type such as equation 2.37 with additional
form factors. The value of the exponent n, depends on the effects included in the
model. For the capillary tube model of Averjanov, with the assumption of a stagnant
air phase was n = 3.5, whereas the extension of the Kozeny model produced n = 3
for an air phase which moves under the same pressure gradient as water (Brutsaert
2000).
For the two phases air and water tortuosity depends on the actual saturation of
the porous medium. At the same value of Sw , tortuosity is typically larger in the
soil water than in the soil air phase and the difference becomes more pronounced
with increasing surface area and at a low Sw . The water phase transport parameters
show a steeper decrease with Sw compared with the air phase parameters, because
the water phase is influenced more by tortuosity than the gaseous phase (Moldrup
et al. 2001).
Pneumatic conductivity is typically described with an effective non-wetting phase
saturation which is complementary to the wetting phase. Examples of these simple polynomial functions are given in Table 2.2. The models do not account for
the existence of a discontinuous air phase, either trapped or locally accessible. To
describe the dependency on saturation the conductivity of a fluid kα (Sα ) (α: water, air,. . . ) is usually written as the product of the conductivity at a matching
point (normally the saturated conductivity ksat,α is used as reference point) and the
relative conductivity kr,α (Sα ).
31
2 Theory of Immiscible Fluids in Porous Media
Model
Corey1
Pirson2 : Imbibition
Pirson: Drainage
Wyllie3
Sw,e = (θw − θw,r )/(θw,s − θw,r )
2
kr,a = (1 − Sw,e )2 (1 − Sw,e
)
2
kr,a = (1 − Sw,e )
1/4 1/2
kr,a = (1 − Sw,e )(1 − Sw,e Sw )1/2
kr,a = (1 − Sw,e )3
Table 2.2: Selected empirical models for relative non-wetting phase conductivity in
terms of effective wetting phase saturation (saturations are defined in
Figure 2.14). In deriving the effective non-wetting phase expressions, the
relationship Sa,e = Sa /Sa,s = 1 − Sw,e was used. Ref.: 1 Corey (1954), 2
Pirson (1958), 3 Wyllie (1962)
The empirical models were developed for the prediction of oil conductivity in
water-wet granular rocks. The model of Wyllie (1962) represents the simple power
models with an exponent of n = 3 which corresponds to the hydraulic Kozeny model.
The Corey (1954) model has an additional term reflecting pore connectivity and tortuosity. The model was widely used by petroleum engineers (Honarpour et al. 1986).
The model of Pirson (1958) reflects hysteresis and offers a separate description of
drainage and imbibition branches of the conductivity-saturation relationship.
2.3.3.2 Geometry Based Conductivity Models
The next step of refinement is to distinguish between portions of pores with different diameters, each with a different conductivity. The pore system is assumed
to be equivalent to a bundle of uniform capillary tubes of many different sizes.
The distribution of pore sizes is derived from the soil water characteristic S(h),
where the matric head h can be related to the effective pore radius r by equation
2.14. The most common parameterizations of hydraulic conductivity functions of
this type are the models of Burdine (1953) and Mualem (1976) which are based
on network models for pores with different features. The refinement of the model
increases the applicability to measured data since influences of pore geometry are
better reflected.
For evaluation of non-wetting phase conductivity, several methods have been developed for the different purposes in soil science, like soil aeration, soil remediation
by air sparging, extraction of volatile organics, or exploitation of natural gas in the
petroleum branch of research. The conceptual framework of the water-air system
and its main features are shown in Figure 2.14; the upper part shows the retention
curve and the lower part shows the corresponding relative conductivity functions for
the wetting and the non-wetting phases. The figure refers to a single branch of the
curve which is typically hysteretic (see Section 2.2.4).
The lower part of Figure 2.14 shows the possible mobility domains of the two
32
2.3 Fluid Flux
h
PSfrag replacements
h
0
1
0
0
Swr
θw,r Sa 1 Sam
θw kra
θw,en 1
θa,en
θw,s
θa
θa,r
θa,s
kr,a
kr,w
Sa 0 Sae 1
Sw
0
Swen
Sws
1
Saen
Sar
0
Sw
krw
1
krw
kra
0
0
Swe
1
Figure 2.14: Schematic representation of retention curve (top) and relative conductivity functions (bottom) for the wetting (dashed) and the non-wetting
(solid line) phases during drainage. In the bottom part the shaded
areas indicate saturations at which one of the phases is discontinuous
(Dury et al. 1999).
33
2 Theory of Immiscible Fluids in Porous Media
immiscible fluids. The lower limit of volumetric phase content achieved by drainage
defines the residual content of this phase (drainage of non-wetting phase is better
known as imbibition of wetting phase). The residual contents θw,r and θa,r are
accounting for the part of the fluid which has lost connection to the phase reservoir
and therefore is incoherently distributed in the sample. If only Darcian flow is
considered, the residual phase content must not be defined rigorously in terms of
thermodynamic equilibria since it represents only an operational concept7 (Luckner
et al. 1989). For purposes of consistent notation, full or maximum phase content of
one phase is defined complementary to the residual phase content of the other phase
as:
θw,s = 1 − θa,r
and θa,s = 1 − θw,r
(2.38)
The Burdine Model
The model assumes a bundle of parallel capillaries with varying pore radii perpendicular to the direction of flow and a uniform pore size in the direction of flow. The
mean hydraulic radius Rw of the wetting phase, is evaluated by integrating the volumetric water content θw from 0 to actual saturation by involving all pores beginning
with the small ones. Combining Laplace equation (2.7) with the hydraulic radius
yields for the wetting fluid:
Z
2
σwa
cos2 γ θw dθw
2
Rw =
(2.39)
θw
p2c
0
For the non-wetting fluid the integration goes from actual saturation to 1 as air
occupies the complementary larger pores:
Z
2
σwa
cos2 γ 1 dθw
2
Ra =
(2.40)
θw,s − θw θw p2c
Surface tension σwa and contact angle γ are assumed to be constant.
With the concept of mean radii, conductivity for the wetting phase becomes:
2
θw R w
kw (θw ) =
τ
(2.41)
and for the non-wetting phase:
2
(θw,s − θw )Rn
ka (θw,s − θw ) =
τ
7
(2.42)
Incoherence only means that the fluid is immobile as a linked phase. Because of trans-phase
exchange (e.g. evaporation, condensation, or degassing) and transport as a dissolved phase in
the other fluid phase, the incoherently distributed phase in the subsurface can still undergo
significant changes.
34
2.3 Fluid Flux
where kw and ka are the conductivities of the wetting and the non-wetting phase at
water content θw , respectively.
Inserting 2.39 in 2.41 and 2.40 in 2.42 yields in:
Z θw
1 dθw
2
2
kw = σwa cos γ
(2.43)
τ p2c
0
and for the non-wetting phase:
Z 1
1 dθw
2
2
ka = σwa cos γ
2
θw τ pc
(2.44)
The relative hydraulic conductivity is:
R θw 1 dθw
kw
0 τ p2
kr,w =
= R 1 1 dθ c
w
k0
2
(2.45)
0 τ pc
and relative pneumatic conductivity is:
R 1 1 dθw
ka
θ τ p2
kr,a =
= R w1 1 dθ c
w
k0
2
(2.46)
0 τ pc
Tortuosity is still in the integral as it strongly depends on the saturation. Burdine
(1953) found an experimental evidence for a linear dependency between τ and θw
because water follows the more tortuous paths as water content decreases:
−2
τ 2 (Sw,e ) = τs2 Sw,e
(2.47)
which led to:
·
θw − θw,r
kr,w (θw ) =
θw,s − θw,r
¸2 R θw dθw
p2
0
R 1 dθwc
0
(2.48)
p2c
for the wetting phase and for the non-wetting phase
·
¸2 R 1 dθw
θw − θw,r
θw p2
kr,a (θw ) = 1 −
R 1 dθwc
θw,s − θw,r
2
0
(2.49)
pc
The Mualem Model
A more statistical approach is to represent the pore system by a bundle of capillaries
with different but uniform radii, cut them normally to the flow direction and rejoin
them again randomly. This model takes the random variations of pore sizes normal
35
2 Theory of Immiscible Fluids in Porous Media
to flow direction and along flow direction into account. The so called statistical pore
space models were invented by Childs and Collis-George (1950).
Mualem and Dagan (1978) deduced a model from this approach where the effective
radius of the whole system is calculated of a serial sequence of two bundles of parallel
capillaries. The geometric mean radius is calculated by:
2
R = r1 · r2
(2.50)
integration over all possible combinations of the two pores yields:
Z Sw,e Z Sw,e
dθ1 dθ2
2
R =
pc,1 pc,2
0
0
(2.51)
Since there is no correlation between the two domains, which was the assumption
for the random rejoining, the integral can be rewritten as:
µZ Sw,e ¶2
dθ
2
R =
(2.52)
pc
0
with the saturation dependent tortuosity
−0.5
τ 2 (Sw,e ) = τs2 Sw,e
(2.53)
the model is in its general form for the wetting phase:
#2
·
¸0.5 " R θw −[1+b]
dθ
p
θw − θw,r
c
w
kr,w (θw ) =
R0 1 −[1+b]
θw,s − θw,r
dθw
pc
(2.54)
0
And for the non-wetting phase:
·
θw − θw,r
kr,a (θw ) = 1 −
θw,s − θw,r
¸0.5 " R 1
−[1+b]
dθw
p
θw c
R 1 −[1+b]
dθw
p
0 c
#2
(2.55)
With this model, which is often used with b = 0, the ka − Sw relationship could
be predicted with a known pc − Sw relation (Mualem 1976). With the examination
of 45 samples, it has been found that the measured data is fitted best with the
exponent 0.5, whereas it has also been proposed by several authors to keep the
exponent as a free parameter for more flexibility in the fitting of measured data
(e.g. Hoffmann-Riem et al. 1999).
Although the refinement of the models reached a better agreement between predicted and measured conductivities, they are based on assumptions which could not
be held true for natural porous media. They are all based on the Young-Laplace
equation, where all geometric influences of the pore space are folded into a single
parameter. The variations of pore size in flow direction implemented in the Mualem
model reflects the natural system better but the assumption of two bundles of capillaries which are connected in series is somewhat arbitrary.
36
2.3 Fluid Flux
2.3.4 Effects Influencing Air Conductivity
Gas flow in porous media differs from liquid flow because of gas compressibility and
pressure dependent conductivity. There is no doubt that the former effect has little
consequences on fluid displacements but it is not always possible to ignore this factor.
Resistance to air flow can be significant during infiltration when compression of air
occurs ahead of a wetting front under border irrigation (Dixon and Linden 1972).
For drainage Corey and Brooks (1975) have found that receding water develops
pressure surges that can be associated with air entry through restrictions within the
soil pore space.
The intrinsic permeability of a porous medium is often measured by means of air
or other gases. Klinkenberg (1941) showed that permeabilities measured by means of
gases vary with pressure and therefore gas conductivities cannot directly be equated
to liquid conductivities. Air passing the porous medium does not act as a true fluid
continuum, which means that the fluid velocity is not zero at the solid boundaries
like it is for other fluids. Because of this effect air conductivity is always larger
than the theoretical values at maximum liquid content. This effect is known as the
”gas slippage” or ”Klinkenberg” effect (Klinkenberg 1941)8 . The difference between
the slip enhanced conductivity and calculations based on intrinsic permeability is
between 20 and 30 percent (Corey 1986).
For fine-textured materials and high gas pressures air conductivity ka has to be
slip corrected by the Klinkenberg method (Klinkenberg 1941):
ka∗ = ka (1 + b/p)
(2.56)
where ka∗ is the slip enhanced conductivity determined at mean absolute pressure
p = (p1 + p2 )/2, ka is the theoretical conductivity in absence of the slip, and b
is a parameter which depends on the geometry of the gas filled pore space. For
simplified models b can be computed from geometric properties, for real soils b must
be determined empirically. For variably saturated media, Fulton (1951) reported a
more or less linear dependence of ka∗ on 1/p, where studies were made on consolidated
media only.
The coherence of the air phase is reflected by the so-called analogy based models.
These models account for discontinuity of the non-wetting phase at a high wetting
phase saturation. The model of Lenhard and Parker (1987) partitions the nonwetting phase into the free and continuous phase Sa,c and trapped or discontinuous
phase Sa,d , i.e., Sa = Sa,c + Sa,d . Other models of this type were developed by
Luckner et al. (1989) and Fischer et al. (1996). The latter will be described in detail
in Section 2.4.
8
When water is used to determine the intrinsic permeability it is smaller than the gas-slippage
corrected air permeability because of the interaction between water and the soil solids, mainly
in clay.
37
2 Theory of Immiscible Fluids in Porous Media
2.3.5 Hysteresis Effects in Conductivity Functions
Capillary-Pressure–Conductivity Hysteresis
The history dependence of the pressure-saturation relation (Figure 2.10), results
in different water contents at a specific capillary pressure during drainage and
imbibition, therefore hysteresis of the retention function affects also the pressureconductivity relation. Hydraulic conductivity between two states of equal pressure
is hysteretic due to the hysteresis in saturation.
Saturation–Conductivity Hysteresis
Hysteresis in the saturation conductivity relation is neglected by most models because the effect is small and models can then be expressed in closed forms (Nielsen
et al. 1986, Kool and Parker 1987).
On the other hand, entrapment of the non-wetting fluid also causes hysteresis in
the relative-conductivity–saturation relation. In the imbibition cycle, water fills the
small pores first which may enclose air in the larger pores. Water has to flow around
these entrapped bubbles and use the smaller pores with low conductivity. The little
differences between drainage and imbibition data was pointed out by Topp and Miller
(1966). Although the observed hysteresis was no more than two or three times the
experimental uncertainty, they found a genuine and systematic effect. Similar effects
were observed by Poulovassilis and Tzimas (1975). They were able to explain the
a
0
rag replacements
a
b
c
0
1
r1
r2
drainage
wetting
r2
b
0
r2
d
w
1
1
1
r1
0
1
r1
0
Figure 2.15: Contour map of the filled pores for the main wetting (a) and drainage
(b) branch (Mualem 1974). (b) shows the drainage map for the saturation corresponding to (a). Hysteresis of conductivity saturation relation
was explained by integration of the different domains for drainage and
saturation.
38
2.3 Fluid Flux
hysteresis effect in their experimental results by the dependent domain theory. The
two effective radii of the cut and rejoined capillary bundle (Section 2.3.3.2) have
a distinct influence during the drainage-imbibition cycle. For a continuous water
phase, all pores corresponding to the applied pressure are filled during the wetting
process. During drainage, the pores with radii r2 are drained if they are in groups
with radii r1 , which correspond to the applied pressure. The integration of equation
2.52 was based on independent domains and the radii of both domains contributed
the same part to saturation. For reflection of hysteresis the domains have to be
integrated separately. The basic principles of this approach are sketched in Figure
2.15.
Lenhard and Parker (1987) have modified the Mualem model and implemented
the concept of phase entrapment in the mobile fluid phases in their empirical model.
The entrapped immobile air bubbles will result in an obstruction to water flow,
while the presence of air bubbles will displace water into larger pores (to reach the
same saturation) which will increase conductivity. Their modified kr − Sw relation
is:
R ∗
R Sa,t dSa,t 2
Sw dSw,e
−
pc (Sw,e )
pc (Sw,e )
0
0.5  0

kr,w = Sw,e
(2.57)
R 1 dSw,e
0 pc (Sw,e )
where Sw∗ is the apparent and Sw,e is the effective water phase saturation. Sa,t is the
entrapped non-wetting phase saturation which can be calculated with knowledge of
the water saturation at the reversal point from main drainage branch ∆ Sw and the
residual air saturation Sa,r . The linear interpolation method of Parker and Lenhard
(1987) described Sa,t depending on Sw as:
dSa,t =
Sa,r
dSw,e
1 −∆ Sw
(2.58)
which results in the hydraulic conductivity relation:
R ∗
2
∗
Sw dSw,e
Sa,r R Sw
dSw,e
− 1−∆ Sw Sw,e pc (Sw,e )
pc (Sw,e )
0.5  0

kr,w = Sw,e
R 1 dSw,e
(2.59)
0 pc (Sw,e )
For air conductivity the effective free air saturation is:
Sa,f = Sa − Sa,t
then the pneumatic conductivity relation is:
"R 1
∗0.5
kr,a = Sa,f
dSw,e #2
pc (Sw,e )
R 1 dSw,e
0 pc (Sw,e )
∗
Sw
(2.60)
39
2 Theory of Immiscible Fluids in Porous Media
Several studies of air conductivity (and other gas conductivity) functions have
considered a drying from complete saturation history followed by a subsequent wetting history (e.g. Honarpour et al. 1986). Most of these studies have obtained lower
gas conductivities for wetting than for drying, given the same total saturation; the
wetting curve often diverges steadily from the drying curve as the wetting-phase saturation increases. Such hysteretic effects were usually ascribed to gas entrapment
(Stonestrom and Rubin 1989b). Previous studies performed by Naar et al. (1972)
and Colonna et al. (1972) have seen the opposite effect. In glass bead experiments
they found that gas-conductivity was higher for wetting than for drying at the same
total gas saturation. For better analysis of conductivity data it is necessary to
know the composition of the air volume in a measured sample, i.e., the portion of
entrapped air, boundary domains, and continuous air-paths, respectively.
2.4 Parameterizations of Hydraulic and Pneumatic
p-k-S–Relations
To obtain a full set of constitutive relations, which describe multiphase processes
in unsaturated porous media the conductivity models 2.48 and 2.54 are combined
with the pressure saturation relation introduced in Section 2.2. A model which describes the hydraulic properties of a porous medium is a combination of the Mualem
approach for conductivity with the van Genuchten parameterization of the retention function. For parameterization of pneumatic properties the geometry based
conductivity models of Mualem and Burdine are also used with the van Genuchten
retention model. Additionally, the emergence point model includes effects of phase
entrapment. Figure 2.16 shows the hydraulic functions (black) and a comparison of
the three different parameterizations of pneumatic properties, the VGM, VGB, and
the EP model which are explained in the following section.
2.4.1 The van Genuchten-Mualem Model — VGM
Van Genuchten combined the Mualem model (equation 2.54) with his retention
model (equation 2.16) and developed a closed parameterization for conductivity
(van Genuchten 1980). This is the most commonly used model for the evaluation of
hydraulic properties in soil physics and also used in this study.
The parameterized model for the wetting phase is:
£
¤2
kr,w (Sw ) = Swτ 1 − (1 − Swn/[n−1] )1−1/n
(2.61)
or in the head form:
kr,w (h) = [1 + [αh]n ]τ [1−1/n] ·
¤2
£
· 1 − [αh]n−1 [1 + [αh]n ]−1+1/n
40
(2.62)
2.4 Parameterizations of Hydraulic and Pneumatic p-k-S–Relations
water
VGM
VGB
EP
relative conductivity [-]
S [-]
log ψ [ cmWC ]
1.0
G
0.8
0.7
theta
nductivity [-]
S [-]
g ψ [ cmWC ]
PSfrag replacements
1.0
0.9
0.6
0.5
0.4
F
0.3
P
0.2
D
0.1
0
0
1
psi
water
VGM
VGB
EP
2
G
0.9
0.8
0.7
0.6
k
PSfrag replacements
0.5
0.4
F
0.3
P
0.2
D
0.1
0
0
1
2
psi
Figure 2.16: Hydraulic and pneumatic functions. Left: Pressure-saturation relation
for water and air. The VGM and VGB models are based on the same
pressure air saturation relation. For EP model effective continuous
air saturation is considering the emergence point saturation. Right:
Pressure-conductivity relations. Separate plots for drainage (solid lines)
and imbibition (dashed lines).
Kool and Parker (1987) have applied their hysteresis approach (Section 2.2.6.3) to
this model in two ways. In order to obtain the most accurate prediction of measured
data they have determined a complete set of parameters for both branches of the
cycle. The model is referred to as the Kool and Parker model. For practical use, it
was assumed that the residual water content θw,r and the saturated water content
θw,s have the same values for drainage and imbibition. Additional simplification was
achieved by neglecting k(θ) hysteresis which meant that parameter n was the same
for drainage and imbibition. The whole hysteresis loop was then described by the
parameter vector (Ks , θw,s , θr , αd , αi , n). Kool and Parker (1987) compared simulation results of this ”constrained” model using six parameters to simulations with
d
i
the whole set of nine parameters(Ks , θw,s
, θw,s
, θrd , θri , αd , αi , nd , ni ). They concluded
that for soils with a wider pore size distribution the reduced number of parameters
would not lead to an unacceptable loss in accuracy. This model is referred to as the
constrained Kool and Parker model.
2.4.2 The van Genuchten-Burdine Model — VGB
For the prediction of unsaturated air conductivity, the Burdine conductivity model
can be combined with the van Genuchten retention model. Insertion of equation 2.16
in 2.49 yields an expression for the relative non-wetting conductivity which depends
on effective water saturation Sw,e and the van Genuchten parameter m
1/m m
)
kr,a = (1 − Sw,e )2 (1 − Sw,e
41
2 Theory of Immiscible Fluids in Porous Media
The second term represents the capillary model, while the first is an empirical factor
introduced to account for connectivity and tortuosity of the pores.
The concept behind the geometry based air conductivity models implies that the
non-wetting phase is continuous up to the maximum wetting phase saturation θw,s .
Non-wetting phase entrapment is considered to be zero, i.e. θa,r = 0.
2.4.3 The Emergence-Point Model — EP
The shaded areas in Figure 2.14 show the saturations where the porous medium
becomes impermeable to one of the phases because residual saturation is reached.
The concept of emergence and extinction points has been established by Stonestrom
and Rubin (1989a) and Fischer et al. (1997). The emergence point is the value of
water saturation where gas flow becomes detectable during drying and the extinction point is the water saturation where the gas flow becomes unmeasurable during
wetting. The range of saturation where both phases are continuous from adjacent
ranges and in which the medium is impermeable to one phase is delimited by these
points (Luckner et al. 1989). This range is a part of the water characteristic that
can be seen as the effective saturation for the wetting phase
Sw,e =
θw − θw,ew
θw,s − θw,ew
(2.63)
and for the non-wetting phase
Sa,e =
θa − θa,ea
θa,s − θa,ea
(2.64)
where θw,ew and θa,ea are the respective phase contents at the emergence point of
the wetting and the non-wetting phases, respectively. θw,s and θa,s are the maximum contents for the both phases. With respect to the wetting phase the difference
between the residual phase content and the emergence point has usually been considered negligible and irrelevant (Brooks and Corey 1964, Corey 1986). In that case,
θw,ew , can be substituted by θw,r so that Sw,e is given by
Sw,e =
θw − θw,r
θw,s − θw,r
(2.65)
As indicated in Figure 2.14 this cannot be done in the same way with the nonwetting phase because of large discrepancies between the residual non-wetting phase
content and the emergence point of the non-wetting phase conductivity (Dury et al.
1998). This effect can be explained by changes in air-filled porosity at saturations
with no air continuity through the sample. The additional air volume increases the
air domains which are in contact with only one of its boundaries (White et al. 1972).
42
2.5 Dynamic Water Transport
This makes a differentiation of the air compartments necessary. Total air content is
the sum of trapped air which is not accessible from the boundaries, locally accessible
air with no contribution to conductivity and continuously distributed domains which
are accessible for air flow measurement. To evaluate experimental air-flow data this
sum of three air compartments has to be considered for the saturation conductivity
relation. As one can imagine, the boundary effect becomes more important for large
laboratory columns where the local accessible air can dominate the volumetric air
content (Dury 1997). If the emergence point is reached, the air conductivity ka is
nonzero and increases with saturation of the air phase. Dependence on saturation
ka (Sa ) is the product of the conductivity kmp = kn (Smp ) at a reference or matching
point Smp and a relative conductivity krn (Sa ). The matching point is the maximum
saturation of the phase which is measured as the residual saturation of the water
phase: Smp = Sa,s . The relative non-wetting phase conductivity is given as
kr,a (Sa ) =
ka (Sa )
ka (Sa,s )
(2.66)
As seen in Figure 2.14, kr,a goes from zero, at a phase saturation below or equal
to the respective emergence point, to one, at the matching point (Luckner et al.
1989).
2.5 Dynamic Water Transport
For isotropic media, the flux equations (equation 2.32) for the two observed fluids
are:
jw = −Kw,sat
∂pw
∂z
and ja = −Ka,sat
∂pa
∂z
(2.67)
Assuming constant density of the fluids, the material balance equations are obtained
by considering a reference volume of bulk medium, including solid matrix as well as
a representative portion of each of the two fluids. The resulting material balance
equations are:
φ
∂Sα
∂jα
=−
∂t
∂z
with α = water; air
(2.68)
The equations of the two phase flow are obtained by the substitution of the flux
equations (2.67) with the expression for permeability (2.33) into the phase conservation equations (2.68):
¶¸
· µ
∂ kw ∂pw
∂Sw
=−
+ ρw · g
(2.69)
φ
∂t
∂z ηw ∂z
43
2 Theory of Immiscible Fluids in Porous Media
and
· µ
¶¸
∂Sa
∂ ka ∂pa
φ
=−
+ ρa · g ,
∂t
∂z ηa ∂z
(2.70)
where the correction for ”gas slippage” has to be accounted for if necessary (Section
2.3.4).
This system of equations is coupled by the conditions (2.6)
pc = pa − pw
and (2.4)
Sw + Sa = 1 where Sw = f (pc ).
If one fluid displaces the other the problem is analyzed by solving equations 2.69 2.70 simultaneously, with appropriate initial and boundary conditions.
In hydrological applications the involved air phase exhibits under natural conditions very small pressure gradients. The air viscosity is two orders of magnitude
smaller than the water phase viscosity9 . In this case the air phase viscosity may
be neglected. Furthermore, the density of air is orders of magnitude smaller than
the density of water. As a consequence, it may be justified to employ a passive air
phase assumption where the air pressure is considered constant. This reduces the
mathematical description of the problem by one equation, i.e. the water-air flow
can be expressed with one single equation. The resulting one-dimensional Richards
(1931) equation is usually written as:
·
¸
∂ψm
∂
∂ψm
Cw (ψm )
=
K(ψm )
− K(ψm )
(2.71)
∂t
∂z
∂z
where the function
Cw (ψm ) =
dθ
dψm
is the specific soil water capacity, and ψm is the matric potential.
9
Material properties for standard conditions (T0 =20 ‰, p0 =101.3kPa) (Lide and Frederikse 1996)
fluid
water
air
44
density [kg/m3 ]
998.2
1.25
dyn. viscosity [Pa·s]
1.002·10−3
1.8·10−5
3 Material and Methods
Direct determination of soil hydraulic properties in steady state experiments is very
time consuming. The pressure saturation measurement can take a long time until
a steady state is reached, especially in the drier range of the measurement where
the hydraulic conductivity is low. To measure unsaturated conductivity at different
pressures it is necessary to establish a constant pressure and flux within the sample,
which is also a difficult task.
A suitable method of determining hydraulic properties, is the dynamic measurement of pressure induced water outflow, with high temporal resolution, in combination with inverse modeling of the underlying transient flow processes. The concept of
the multistep outflow approach which was used for this study is described in Section
3.1. The experimental setup and the measurement procedure which provides necessary information about cumulative outflow of water, tensiometer potential, and
air conductivity is described in Section 3.2. The preparation of the samples and
settings of the technical equipment for a combined measurement of water and air
dynamics is described in Sections 3.3 and 3.4, respectively.
Estimation of hydraulic properties with the inverse modeling tool eshpim is described in Section 3.5. The method for the evaluation of measured air conductivity
data is introduced in Section 3.6.
3.1 Multistep Outflow Method
Measurement of transient outflow data and the inverse approach of solving the onedimensional Richards equation numerically began in the 1980’s. Kool et al. (1985)
made onestep outflow experiments and compared their parameter estimations with
steady state measurements. This method required additional independent measurements of soil water content data (van Dam et al. 1992) or matric head data (Toorman et al. 1992) in the optimization procedure to reduce problems of uniqueness.
These problems were circumvented with the multistep outflow method invented by
van Dam et al. (1994). Pressure had been changed in several small steps and the
response of the system in stages provided sufficient information in the multistep
outflow data, for a unique estimate of hydraulic functions. In the following several
improvements were made with the multistep outflow method including automated
45
3 Material and Methods
0.30
10
5
ag replacements
PSfrag replacements
2
theta
0.25
0.10
0
Ausfluss
-5
0.05
0
Druck
utflow [cm]
cumulative outflow [cm]
time [h]
time [h]
re [ cmWC ] lower boundary pressure [ cmWC ]
ψ [ cmWC ])
log (ψ [ cmWC ])
vity [cm/h]
conductivity [cm/h]
θw [-]
θw [-]
2
150
-15
-20
1
psi
-10
1
0.20
0.15
k
100
50
0
0
5
10
15
t [h]
20
25
0
1
psi
2
Figure 3.1: Left: Multistep outflow measurement curve. Lower boundary pressure
(dashed, right axes), cumulative outflow, and tensiometer measurements
(symbols) together with respective simulation results (solid lines). Right:
Hysteretic hydraulic functions as evaluated by inverse simulation from
the drainage (solid lines) and the imbibition (dashed) branches of the
outflow curve. Top: Pressure saturation relation. Bottom: Conductivity
pressure relation.
matric head measurement, or additional moisture measurement by TDR, or gamma
attenuation, which increased the uniqueness of the parameter values (Hopmans et al.
2003). The combination of cumulative outflow data with tensiometric data in the
objective function improved the method by producing more accurate estimations of
the hydraulic functions for a wide range of soil textures. The problem with this
method lies in the description of underlying process models. The unsaturated conductivity functions can only be estimated with a satisfactory validity if the retention
model is able to describe the relation (Durner et al. 1999).
Figure 3.1 (left) shows an example of a typical multistep outflow experiment. The
examined sample was a coarse-textured sand column with a height of 10 cm. Starting from full water saturation, the water phase pressure was decreased in stages
as shown in the figure by the dashed line. The symbols (◦: cumulative outflow,
4: tensiometric potential) show the response of the system on the change of lower
boundary pressure where the outflow was measured in cm3 /cm2 . Information about
air entry pressure was provided by the beginning of outflow. The slope of the θ(ψ)
distribution was provided by the height of the outflow steps in corelation to the
applied pressures. The height of the distinct pressure steps played a crucial role for
the estimation of the retention characteristic. Hydraulic conductivity could be estimated from the shape of the outflow steps and the difference between saturated and
residual water content could be estimated from the maximum value of cumulative
outflow.
46
3.2 Experimental Setup for Multiphase Measurements
Results of the evaluation of hydraulic parameters by one-dimensional inverse simulation are shown in the same figure with the solid lines representing outflow curve and
tensiometric potentials, respectively. Hysteretic retention and conductivity curves
on which the simulation was based is shown in the right part of Figure 3.1.
3.2 Experimental Setup for Multiphase
Measurements
The apparatus presented in Figure 3.2 consists of three major units, a control unit
and two separate units for the measurement of water flow and air dynamics. The
control unit is represented in the figure by a computer and vacuum containers. The
computer is used for logging and storing pressure sensor data and for switching of
the magnetic valves. The vacuum containers provide the low pressures, necessary
for the experimentation. The multistep outflow measurement device is shown on
the left hand side of the figure while the air-flow measurement device is presented
on the right hand side. The sample holder is located in the upper central region of
the figure.
3.2.1 Mulitstep Outflow Measurement Device
The multistep outflow part of the experimental setup was based on the classical
arrangement of the sample and the measurement device. The column with a maximum diameter of 168 mm was placed vertically on a porous plate. System disturbances which induce transient out-/inflow were applied via water phase pressure
at the lower boundary of the sample. The custom method for the lower boundary
connection, applying a thin nylon membrane with a high air entry value and low
resistance to the water phase could not be used here. For additional examination
of air-flow behavior in the hydraulic and pneumatic measurements, the fluid phases
had to be separated. Therefore, a segmented porous sintered glass plate was used
which is described in detail in Section 3.2.3. For the hydraulic measurement, the
hydrophilic sintered glass part of this plate was used as interface between the sample
and measurement devices. The glass had an air entry pressure much lower than the
measurement range during the experiments, i.e. it had been water saturated during
the whole experiment.
The left part of Figure 3.2 shows the device for the determination of hydraulic
properties (water-flow). The lower boundary pressure was mixed in a pressure barrel
by switching the valves V1 which was connected to a −500 hPa vacuum reservoir and
V2 which was opened to atmosphere. The lower boundary pressure was monitored
by the pressure sensor P S1 and the software adjusted the pressure by switching the
corresponding valve.
47
3 Material and Methods
L6
L5
L12
L7
L8
g replacements
L23
burette
ubble tower
L15
L22
PS1
PS2
PS3
PS4
water flow
air-flow
sample
ensiometer
soil air
V1
V2
V3
V4
V5
V6
outflow
reservoir
vacuum
lift
pa
L3
L1
L13
L9 L21
L4
L20
L14
L2
L16
L17
L11
L19
L10
L18
Figure 3.2: Setup for simultaneous measurement of water and air dynamics. The
sample (center) is mounted on a phase separator (see Figure 3.4). Water
flow (left) is measured by the multistep outflow method. Water pressure at lower boundary is controlled by magnetic valves V1 and V2 with
feedback from pressure sensor P S1 . Cumulative outflow is collected in
a burette and monitored by sensor P S2 . Air-flow measurement (right)
is conducted by collecting the cumulative outflow in a Mariotte bubble
tower. Air pressure is controlled by height of outlet which is adjusted
by the lift, the applied pressure is measured by sensor P S3 . Cumulative
air volume is measured by sensor P S4 . Valves V3 -V5 are used to refill
the bubble tower. V6 is for drainage of the air grid within the separator
plate.
48
3.2 Experimental Setup for Multiphase Measurements
For the experiment, the water pressure was changed in single steps at the lower
boundary and the corresponding cumulative outflow was collected in the burette
where the water table was monitored by the differential pressure sensor P S2 and
logged with high temporal resolution by the computer.
3.2.2 Mini-Tensiometers
To improve the parameter estimation with the inverse procedure, the tensiometer
potential was measured during the multistep outflow measurements. With this
method the speed of the spreading pressure front and the actual capillary pressure
at a specific position could be measured. As some of the materials were rigid, it
was not possible to use custom tensiometers, which have to be placed inside the
samples. The special external tensiometer designed for measurement with rigid
media is sketched in Figure 3.3. This instrument consists of a slice of sintered glass,
thickness 4 mm, with a high air entry value (> 150 cmWC ). Conductivity of this
material is 0.4 cm/h. The membrane has a rigid plastic tube casing. The tip had
to be as flat as the surface of the sample on which it was placed, for maximum
contact. Contact area could be improved by some contact material (fine sand, gaze,
. . . ) and an additional rubber casing prevented the tensiometers from drying up
during a measurement. For external use, the tensiometer was adjusted properly at
the column surface and tightened in a chuck to maximize hydraulic contact.
With the setting used the pressure sensor measured capillary pressure as the
difference between water pressure and atmospheric air pressure outside the column.
For internal use of tensiometers in the sand packings it had to be taken into account
that air pressure of entrapped air might be higher than atmospheric air pressure and
the tensiometer might only measure water pressure and not capillary pressure.
3.2.3 Air-Flow Measurement Device
Current methods of multistep outflow analysis are based on the basic assumptions
that gas and water flow in the porous medium can be decoupled as the governing
Richards equation (2.69), on which the inversion is based, includes the hypothesis
that the gas phase in soil is always at atmospheric pressure (Section 2.5). Since the
air phase viscosity is much larger than water phase viscosity, ηa > ηw 1 , air should
be at atmospheric pressure throughout the unsaturated medium.
As mentioned above, the aim of the device designed for this study was a simultaneous measurement of hydraulic and pneumatic properties of the laboratory sized soil
columns. To detect a continuous air phase through the sample, a vertical setting of
air-flow measurement was chosen. Direction of air-flow was top-down where air was
extracted at the lower boundary which reduced phase interactions to a minimum.
1
For the material parameters at standard conditions see footnote on page 44
49
3 Material and Methods
L3
L2
PSfrag replacements
5 mm
20 mm
pressure sensor
L1
Figure 3.3: External tensiometer designed for use with rigid porous media. The
porous sintered glass plate (light grey) connects to the pressure transducer via a firm plastic tube. The rubber band (magenta) is used for
sealing.
With the additional information about air continuity and air conductivity along the
z-axis the saturation range at which Richards equation is valid for the entire column
could be ascertained.
Phase Separation
For simultaneous measurement of hydraulic and pneumatic properties of the porous
medium in a single experiment there was a need to separate the involved phases
(solid, water, and gas). For this purpose a phase separator for the lower boundary
was designed which allowed the control of the boundary conditions for air- and waterpressure and provided an interface for further devices to measure air- and water-flow
(Figure 3.4). The vertical arrangement of the sample had the disadvantage that the
whole weight of the sample was loaded at the separating interface. As mechanical
stability was required on the one hand and conductivity of the plate had to be
maximized on the other, a sintered glass plate of 10 mm thickness was constructed.
This plate was supported by an underlying PVC grid which was part of the plate
holder2 .
For the separation of fluids, wetting properties and air entry pressures of different porous materials were utilized. The sintered glass with an air entry pressure
of −500 cmWC was the hydrophilic part (small contact angle). Water percolating
2
First test with thinner plates and a different grid geometry always caused the plate to crack when
a column was placed on top and additional to this a pressure was applied. See also Appendix
A.2
50
3.2 Experimental Setup for Multiphase Measurements
PSfrag
L1
replacements 10 mm top view
cross section
200 mm
L2
L5
L5 L3
L4
Figure 3.4: Porous plate to separate air and water at the lower boundary: hydrophilic sintered glass plate (grey), hydrophobic sintered HDPE (red),
and air channel (light grey) to facilitate air-flow.
sintered glass
pore diameters [µm]
5-8
◦
contact angle [ ]
<10
porosity [%]
48
conductivity [cm/h]
18
HDPE
500-1000
160
52
>500
Table 3.1: Physical properties of the two materials used for separation of the fluids
at the lower boundary.
through this plate was collected in a funnel and passed to the burette where it was
gathered cumulatively. As a path for extraction of soil air, 3 mm deep channels were
milled into the glass plate and covered with a 1.5 mm
soilair
mariotte
layer of sintered hyper-dense polyethylene (HDPE), a
PSfrag
replacements
hydrophobic material (large contact angle) with pores
porous
> 500 µm. The 1.5 mm channel below the HDPE-grid
plate
was an air conduit which avoided distance effects within
porous
membrane
the plate, and improved air conductivity inside the
soilde-air
vice. The physical properties of the two materials
used
to burette
burette
for phase separation are summarized to
in Table
3.1.
bubble tower
A major hurdle with the separation of the phases was Figure 3.5: Additional
the flow of water through the air channels caused by thin
separator
films or spontaneous movement of small drops. Since it
was not possible to avoid a minimal flow of water through the air outlet of the
51
3 Material and Methods
sample holder an additional water separator (Figure 3.5) was installed directly at
the air outlet of the sample holder. The chamber with the air in- and outlet at the
upper end had a porous membrane, 3 mm thick, at the bottom which was in contact
with the water collecting burette. Water pressure in this separator was the same as
the pressure applied to the sample. Any water leaving the sample at the air outlet
was added to the cumulative outflow burette by this construction.
Measurement of Air Volume
The determination of air conductivity in laboratory experiments required a highly
precise and flexible instrument for the observation of flow rates. Flow chambers or
soap film permeameters are typically used for this purpose, as they offered recommended precision (e.g. Wu et al. (1998), Moldrup et al. (1998), Corey (1986)), but
since these are not automatically operated, a necessity for the experiments which
lasted for many weeks, they were of no practical use.
The most adequate instrument which met all requirements was the Mariotte bubble tower. With this instrument, flow measurement of high resolution in time and
volume could be made. The air pressure, adjusted by a lift (Figure 3.2, right), could
continuously be measured by pressure sensor PS 3 . Also by this method, a very
small and permanent air pressure difference of 0 - 5 hPa could be applied with a precision of ±0.05 hPa. The simple concept along with the static settings of the bubble
tower made it a very robust instrument for air volume counting. Resolution of the
measured volume was influenced by both the response time of the sensor and the
width of the used air outlet which shapes the bubbles to a specific discrete volume.
The outlet diameter of 5 mm, was the best compromise between both the intrinsic
conductivity of the instrument and resolution of measurement.
The air influx collected in the bubble tower where the height of the water table
was monitored via differential pressure sensor P S4 . Although this technique was
limited by the volume of water in the tower it was the most flexible method for air
conductivity measurements. The burette could collect about 900 cm3 of air. After
this amount had passed through the soil column the burette had to be refilled.
The valves V3 - V5 were switched and the vacuum sucked the water back into the
tower. The refill interruption took about 5 minutes, and did not affect the air-flow
measurement. If necessary, a second tower could be installed and they could both
be used in an alternating mode without interruption.
If the gravitational component of the driving force is negligible compared to
the force resulting from the applied pressure gradient, then the one-dimensional
isothermal flow of air can ideally be described by the extended Darcy equation for
gases
ja =
ka dpa
ηa dz
(3.1)
where ja is the volumetric flux, ηa is viscosity, dpa /dz is the pressure gradient in
52
3.2 Experimental Setup for Multiphase Measurements
PSfrag replacements
5
14
3
0
12
2
-5
10
1
-10
8
4
mariotte
40
150
30
airpres
200
tower
50
aircond
0
100
20
6
-20
4
-25
2
-30
0
airbound
10
-2
0
-15
outflow
-1
50
out
airperm
bc
250
0
onductivity [cm/h]
60
bcp
water-flow
time
ative outflow [cm]
bcair
bubble tower [cm]
bcair [hPa]
bcwater
bcwater [ cmWC ]
ow (bubble tower)
300
-3
0
1
2
4
3
5
6
7
t
Figure 3.6: Example of an air-flow measurement. Blue: Lower boundary pressure
(water). ◦: Cumulative outflow of water. Red: ×: Measured height in
Mariotte bubble tower; 3: Lower boundary pressure (air). Solid line:
Effective air conductivity.
z-direction, and ka is air permeability. Because of the laminar flow, a correction for
the compressibility of the gas is not required.
A correction due to the Klinkenberg Effect (Section 2.3.4) was not necessary for
this study because the pore diameters were large enough and the applied air pressure gradients were small. Several experiments, with various air pressures gradients
applied to coarse and fine-textured columns, found linear dependence between pressure and volume, i.e. ki was constant and there was no influence of gas slippage
which corresponded to the results of a study by Wu et al. (1998).
Figure 3.6 shows an example of a combined multistep outflow and air conductivity
measurement. Hydraulics are represented by the blue dashed line which is the
lower boundary pressure applied to the water phase3 and the black circles which
denote the cumulative outflow4 . Pneumatic measurements are represented by the
lower boundary pressure applied to the air phase5 (red diamonds), the air volume
cumulated in the bubble tower6 (red crosses), and the solid green line gives the
effective air conductivity calculated with equation 3.1.
The air-flow measurement started at a water pressure less than 0 cmWC 7 . The
3
measured with pressure sensor P S1 (see Figure 3.2)
measured with differential pressure sensor P S2 (see Figure 3.2)
5
measured with pressure sensor P S3 (see Figure 3.2)
6
measured with differential pressure sensor P S4 (see Figure 3.2)
7
With the vertical arrangement of the setup and the separator at the lower boundary it is not
possible to measure air-flow at water pressures higher than 0 cmWC because the air grid is
4
53
3 Material and Methods
air grid was drained by opening valve V6 for 5 minutes (Figure 3.2). The water
which was collected in the burette was not added to the cumulative outflow to
avoid misinterpretation of hydraulic data. For air-flow measurement valve V4 was
opened. Lower boundary air pressure was kept stable during the whole period
of measurement. Interruptions in the lower boundary air pressure occurred if the
bubble tower had to be refilled (after 5.5 and 6.5 hours).
With the experimental setup described above, it was possible to measure an integrated air-flow through the sample. To get air-conductivity relations which were
corrected for all influences of vertical saturation gradients due to gravity effects this
measurement had to be evaluated by concerning these effects. The method of data
analysis is described in Section 3.6.
3.3 Preparation of the Samples
For the measurement of multiphase properties it was absolutely essential to have a
well prepared sample. Any open voids between the sample and the measurement
device can have unpleasant consequences. In the case of hydraulic measurement gaps
could lead to capillary barriers during imbibition which could have an impact on the
estimation of hydraulic conductivity. In the case of the measurement of pneumatic
properties, air-flow leaks along the boundaries prepared a kind of shortcut, which
increased air conductivity. Air inclusions in the primary drainage cycle could also
lead to measurement errors.
Sealing
Boundary effects were reduced by the sealing of the samples with special care for
all possible gaps at each boundary. Several wrapping and sealing techniques were
tested on the rigid porous media. Latex casings had the advantage of being able to
follow the shape of the columns. However, for fully saturated samples the elastic
membrane gave way to the pressure and additional water was found inside the casing
but outside the sample. This effect led to errors in maximum water saturation.
Silicone gaskets influenced the surface of the glass beads. Wettability of a column
wrapped in silicone was reduced at the boundaries. A similar effect was reported by
Corey (1986), where an extremely small contamination with silicone-based lubricant
for stopcocks, prevented the formation of stable soap films in a flow meter and was
extremely difficult to remove. Best results were achieved for the rigid materials with
shrinking PVC tubes, a product normally used in the electronic field. A tube was
slipped over each sample and heated until shrunken to the minimum size. The warm
tube adapted to the shape of the sample surface and the gaps were closed.
Installation of the wrapped sample in the sample holder also required waterand air-tight boundaries. For this purpose the rigid columns were first sealed in
saturated with water then.
54
3.3 Preparation of the Samples
the sample holder then evacuated in the desiccator and finally mounted on the
separator plate. Effects of initial air inclusions and boundary gaps were removed by
this procedure.
Sealing problems did not occur with the sand packings as they were compiled
directly in the sample holder. The flexible sand fitted the sample holders geometry and boundary gaps were possible only within the grain size of the respective
sand.
Saturation
For examination of primary drainage curves all measurements had to start in a fully
water saturated state. The method of slow capillary rise led to insufficient results
as there was still entrapped air in the samples. For the rigid samples, capillary rise
led to 90 % water saturation only. The remaining air was removed by saturation in
a desiccator. With this method air was totally removed from the sample. Different
durations and pressures were tested and best results were achieved when samples
were kept for 24 hours in the desiccator at a pressure of −500 cmWC .
This type of saturation did not work for the nonrigid sand. In every experiment the
dry sand was poured into the water-filled sample holder, which led to full saturation.
As the sand took the shape of the sample holder, and since neither swelling nor
shrinking occurred, no extra sealing was necessary.
Hydraulic Contact
The main disadvantage of rigid materials was the hydraulic contact at the interface
between the sample and the measurement device. Although the sample bases and
the porous separator plate were grind flat, installation of the samples in the holder
would lead to small vertical rotations which would cause gaps at the interface in
the range of some hundred micrometers. These gaps made data analysis impossible
because conductivity of the whole system was disturbed, especially in the imbibition
part of the measurement. The problem was solved by adding a thin cotton mesh
as an immersion layer. The elastic mesh was able to close gaps and allowed water
and air to pass in their distinct compartments of the separator plate. The influence
on the hydraulic and pneumatic conductivity was taken into account as saturated
conductivity of the separator plate was measured with the mesh installed.
Tensiometers were also connected to the sample via cotton meshes when they were
used in the external mode. For internal use of tensiometers in the sand packings it
was necessary to avoid gaps between the instrument and the sample. These types of
gaps could lead to a loss of hydraulic contact between the instrument and the sample
and therefore distort air conductivity measurements by offering air channels. After
installing the tensiometer water was poured into the area around the instrument
which made the sand flow through the gaps and close them.
55
3 Material and Methods
L7
L1
L2
L6
PSfrag replacements
pressure
sensor
sample
free drainage
0
l
zw (t)
L3
L5
L4
Figure 3.7: Falling head permeameter for the determination of saturated hydraulic
conductivity (Klute and Dirksen 1986).
Saturated Conductivity
For modeling of the experimental results, saturated hydraulic conductivity of the
plate
separator plate had to be included in simulations. Kw,sat
was measured with the
”falling head method” (Figure 3.7, Klute and Dirksen 1986). The plate was ponded
with water and during free drainage the water-level zw (t) was monitored by a pressure sensor. With this information the saturated conductivity of the plate could be
calculated using Darcy’s equation:
plate
Kw,sat
= − l · t · log
zw (t)
z0
(3.2)
with the initial height of the water table z0 , thickness of the plate l, and duration of
drainage t. The measured values which were used for simulations of measured data
are listed in Table 3.1.
The measurement of saturated hydraulic conductivity of the separator plate was
repeated after every experiment, because small particles could have been eroded
56
3.4 Experimental Settings
in the sample and settled down on the plate during the measurement. With the
materials used no change in saturated conductivity of the plate was observed.
The falling head method was also applied to measure the saturated conductivity
of the examined samples. The rigid columns were placed in a tube and ponded
with water, the sand packings were measured together with the separator plate.
Conductivity of the plate and the sample were separated by
sample
Ksat
=
lsample
lsample +lplate
l
− Kplate
plate
K total
sat
(3.3)
Temperature and Humidity
Measurement of hydraulic and pneumatic properties depended on the temperature
and humidity of the surrounding area. Influences were kept small by a climatic locker
where these parameters could be held in stable values. Because of interferences
between the locker’s electronics and the measurement electronics the locker was
used to establish a homogeneous humidity (almost 100 % rel. humidity) at room
temperature (varied between 23 - 26 ‰) and was shut off afterwards. While the
experiments were running it worked as insulation. The premoistened air prevented
drying of the samples during air-flow measurement.
3.4 Experimental Settings
Calibration of Pressure Sensors
All pressure sensors involved in the experiments were calibrated every time they
were used (for technical details of the sensors see Appendix A.1). Linearity between
the applied pressure and the digital output signal was constant over a long period of
time (> 1/2 year) but the offset could change within a few weeks. Because of this
effect, calibration was done before and after every measurement. In theory, the drift
of the offset could be corrected if there was measured data parallel to the drifted;
otherwise the experiment would have to be repeated. All pressure sensors worked
with an absolute precision of 0.1 cmWC .
Software Handling
Realization of the simultaneous multistep outflow and air-flow measurement and
evaluation of the measured data required several software tools. To make the handling of all used software easier, they were merged in an easy to use front-end. This
graphical user interface (GUI) was developed in LabView (Instruments 2001) which
is a graphical programming language that offers special features for user dialogs,
data processing, and online data visualization. With this GUI, all input file editing which is necessary for the software and which controls the experiment can be
prepared stepwise:
57
3 Material and Methods
50
3
"channel 4"
"channel 16"
"channel 3"
5
outflow data
outflow fit
tens data
tens fit
2.5
40
0
2
30
20
-5
10
1.5
-10
0
1
-15
-10
0.5
-20
-20
0
-30
0
-0.5
10000
20000
30000
40000
50000
60000
70000
80000
90000
100000
-25
0
5
10
15
20
25
30
Figure 3.8: Graphical user interface for setup, control and evaluation of multistep
outflow experiments. The tab structure allows only a small insight to all
available features.
ˆ calibration of all sensors used (absolute/differential)
ˆ administration of available sensors
ˆ controlling of sensors and magnetic valves
ˆ boundary condition settings for the multistep outflow and air-flow measurement
ˆ logging and online visualization of pressure sensor data
ˆ evaluation of hydraulic parameters with the eshpim program
ˆ visualization of fitting results and hydraulic functions
58
3.4 Experimental Settings
Figure 3.8 shows a part of the GUI which is arranged in a tab structure8 . With
this panel configuration, termination, supervision, and evaluation of the experiment
could be done in an intuitive convenient way while the specific tools were working
in the background.
Boundary Conditions
A typical transient water flow experiment started with the water-saturated column.
Pressure at its lower end was reduced in a series of steps to some minimal value,
then increased again through a series of steps to its starting value. The experiment
was controlled and monitored by a computer with a highly temporal resolution, i.e.
all pressures were recorded every ten to sixty seconds. The typical duration of a
multistep outflow experiment was about 40 - 60 hours for the main drainage and
imbibition curves.
The additional air-flow measurement had to be started at a lower boundary water
pressure < 0 cmWC . After the drainage of the grid, the two phases could be separated
at the lower boundary and the flow of each fluid could be measured in the designated
parts of the apparatus.
Application of an air pressure gradient could have lead to the redistribution of
water during the air conductivity measurements. Large air pressure gradients could
also have burst some of the water films blocking pores to gas movement (Ball et al.
1981) and the Reynolds number could have become large enough for laminar flow
to begin changing to turbulent flow, where Darcy’s law no longer holds. The effect
should have been minimized by using only air pressure differences up to 5 hPa. But
instead, a test of several measurement modes showed that the best results were
achieved with a selective mode at specific points correlating to the water outflow
curve. To reduce any kind of influence on the hydraulic measurement, air conductivity was measured at the beginning of a hydraulic pressure step after the fast rise
of the outflow curve, and depending on duration, at several additional times during
the redistribution phase of the step. If the phases were separated properly, both
measurements should not have influenced each other9 .
In the rewetting phase the air-flow measurement was stopped at a lower boundary pressure of 0 cmWC which prevented a rise of the water-table higher than the
separator plate.
8
The whole GUI was organized into three levels, general settings, experiment type settings and
experimental settings. All three levels had different panels where specific features could be set.
The different panels could be accessed by the tabs of the panels.
9
Any disturbances of the hydraulic data would have shown an inadequate separation and would
have made the measurement worthless.
59
3 Material and Methods
3.5 Estimation of Hydraulic Parameters
The inverse modeling approach estimates soil hydraulic properties from transient
experiments, giving much more flexibility in experimental boundary conditions than
required for steady state methods. Scanning of hydraulic state variables, θ and ψ,
can be done continuously over a wide range of pressures. With this approach both
the soil water retention and the unsaturated conductivity function can be estimated
from a single transient experiment. The governing equation for this process which
is used for simulations is the Richards equation. Evaluation of the dynamic water
flow experiments was done with the inverse simulation code eshpim (estimation of
soil hydraulic properties by inverse modeling), which was developed by Zurmühl
(1996). Using the lower boundary conditions, the cumulative outflow data and the
tensiometer potential, hydraulic parameters were determined by a combination of
solving Richards equation (equation 2.69) numerically (direct problem) and by a
method for solving the minimization problem for the parameters used (equation
2.15). A brief summary of the parameter estimation of eshpim is given in the
following section.
Direct Problem
Simulations of the direct problem were based on the one-dimensional Richards equation in the mixed form (Celia et al. 1990):
µ
¶
∂θ
∂
∂ψ
=
K(θ)
− K(θ)
(3.4)
∂t
∂z
∂z
where θ is the volumetric water content, t is time, z is depth, K(θ) is hydraulic
conductivity and ψ is the matric head. For hydraulic functions the van Genuchten
– Mualem approach with their respective parameterizations were used.
Additionally initial and boundary conditions as used in the experiment had to be
specified. Finite differences simulations were done for n discrete nodes of spacial
discretization and a fully implicit method for temporal discretization.
Inverse Problem
The parameter set was improved iteratively until the deviation between measured
and calculated values was minimal. The target function which had to be minimized
by the least squares method is:
χ2 (η) = 1/2 [y − f (x, η)]T W [y − f (x, η)]
with
60
η
o
χ2 (η)
y
f (x, η)
parameter vector, η = (η1 , η2 , . . . , ηo )T , (T : transposed)
number of parameters
sum of least squares
vector of measured values yi , i = 1, . . . , l;y = (y1 , y2 , . . . , yl )
calculated values
(3.5)
3.5 Estimation of Hydraulic Parameters
x
W
l × k-matrix with k independent variables (time, space)
and l values
l × l-matrix with weighing factors
Kool et al. (1985) developed a method for the use of more data sources with specific weighing factors. With this approach all measured data (outflow, tensiometer
potential, saturations, . . . ) could be used for optimization.
To solve the problem, eshpim used the Levenberg-Marquardt algorithm. Equation
3.5 was minimized by finding the zero point of the derivative related to η. This
resulted in one equation for each parameter ηi . As the parameterization used was
nonlinear, which lead to a nonlinear function in η, the system of equations was
solved iteratively until a minimum was reached.
Mulitmodal Pore Systems
For structured soils, the water retention function is not always represented well by
a unimodal model, if pore size distribution has additional maxima in the range of
large pores. Several concepts of data evaluation have been developed during the
last years which were able to describe multimodal pore systems (Ross and Smetten
1993, Durner 1994, Zurmühl and Durner 1998). The model implemented in the
eshpim code is the one developed by Durner (1994) which describes the multimodal
retention curves by a superposition of several van Genuchten curves:
k
X
θ − θw,r
S(ψm ) =
=
wi (1 + (αi ψm )ni )mi
θw,s − θw,r
i=1
with
k
wi
for ψ < 0
(3.6)
number of different modes
P
weight of single modes with 0 < wi < 1 and
wi = 1
This type of multimodality assumes that the different pore systems are distributed
homogeneously over the whole sample. This means that they have to be in contact
with both the air and the water reservoir to show a simultaneous response. In the
eshpim code a bimodal model is implemented which allows to distinguish between
two different maxima in the pore-size characteristic.
Hysteresis Model
The hysteresis model implemented in eshpim was the constrained Kool and Parker
(1987) model (Section 2.4.1). The model was based on main drainage and imbibition
curve with known parameters. If wetting direction was changed from drainage to
imbibition, θr of the drainage curve was changed to get a closed loop of drainage
and imbibition retention curves. Since hysteresis in the K(θ)-relation is insignificant
for most problems (Topp 1969) it was neglected by eshpim. This constrained type
61
3 Material and Methods
of the Kool and Parker model resulted in a single parameter α2 for the imbibition
branch which includes all hysteresis effects.
For fine grained homogeneous materials, the described hysteresis model was not
sufficient. To get more flexibility in the hydraulic functions for both branches of
the multistep outflow experiment, fitting of data was split into two separate parts.
For this purpose hysteresis in the K(θ)-relation was assumed. This model has been
introduced as the Kool and Parker model in Section 2.4.1. The outflow branch
was fitted with a complete set of parameters to an optimum agreement. For this
set of parameters, tension and water contents were simulated for the reversal point
where the lower boundary pressure changed from drainage to imbibition. These
simulation results were used as initial conditions for the simulations of the imbibition
curve. As the columns response to the change from drainage to imbibition was
delayed, measured data could not be used for fitting until the change was carried out.
Estimation of hydraulic properties were done with a full set of parameters for the
second branch of the multistep outflow curve which led to a better correspondence
between measurements and simulations.
Statistical Characteristics
During the parameter optimization procedure, statistical characteristics for fitted
parameters were calculated. The correlation of parameters was evaluated by the
correlation matrix:
cij = bij (bii bjj )−0.5
with
bij
Jw
J
S
(3.7)
elements of matrix B = (JTw Jw )−1
SJ
Jacobian matrix
Cholesky splitting of matrix W
where high correlation of two parameters meant that there was not enough measured data for independent evaluation of both parameters or there was a physical
correlation which was not taken into consideration by the underlying model.
Additionally the confidence intervals for all parameters were calculated by:
2
0.5 2
ηj − b0.5
jj s t1−α/2 (n − p) ≤ η̂j ≤ ηj + bjj s t1−α/2 (n − p)
with
62
ηj
calculated parameter value
η̂j
’apparent’ parameter value
t1−a/2 (n − p) value of t distribution for confidence number 1 − α
with n − p degrees of freedom
rT
2
w rw
s = n−p
variance in error
(3.8)
3.6 Calculation of Air Conductivity
As the correlation matrix and the confidence intervals were found by linear regression
methods they did not really reflect the true values for the nonlinear system. The
confidence interval should have been seen as a confidence region. All statistic results
were only valid if the calculated parameter vector was the ’true’ minimum (for more
details about the eshpim parameter estimation program see Zurmühl 1994, Zurmühl
1996).
3.6 Calculation of Air Conductivity
The calculation of air conductivity based on the measurement which was introduced
in this chapter had to consider the fact of a water saturation gradient within the
sample due to the transient conditions of the experiment and due to gravity effects
during the stationary periods of the experiment. Measured data, which reflected an
integral response of the entire system had to be compared to locally and temporally
discretisized simulations. With simulated local properties, global properties could
be calculated and compared with measured data.
Simulation of Water Dynamics
Evaluation of measured air permeability data required knowledge of the part of
pore space available for the gaseous phase. Simulation of water dynamics therefore resulted in a vertical discrete distribution of water content within the sample.
Hydraulic functions evaluated by inverse modeling were used for one dimensional
simulations and the direct problem was solved as described in Section 3.5. Spatial resolution of the sample height was 0.05 - 0.2 mm. Temporal resolution followed
closely the times of measured data recorded with the multistep outflow experiments.
A result of this modeling was the temporal evolution of vertical profiles of local water
saturations (Figure 3.9 center).
Simulation of Air Conductivity
The simulated water saturation profiles were used to calculate effective pneumatic
conductivity over the whole height of the columns. For the three models introduced
in Section 2.3.4, effective air saturation and the resulting air conductivity were
calculated for a discrete one-dimensional sample. Air conductivity of the entire
column was calculated with:
H
1
= P h∗
ka,c
n ka,n
where ka,c
H
h∗
ka,n
n
(3.9)
effective air conductivity of the column
height of the column
height of an element
effective air conductivity of element n
number of elements
63
3 Material and Methods
Modeling of local pneumatic properties was based on the effective hydraulic parameters estimated from the multistep outflow experiments. For heterogeneous samples
pneumatic properties were calculated based on the material properties of the homogeneous components used for assembling. For simulations of the structured samples
the bimodal approach was used as proposed by Zurmühl and Durner (1998).
The pneumatic conductivity models VGM, VGB, and EP introduced in Section
2.4 were chosen as they are based on three different concepts for the pore space which
was available for air phase transport. The van Genuchten-Burdine model (VGB),
was based on the parallel capillary approach with the empirical shape parameter, is
supposed to reflect the geometry of the hydraulic but not the pneumatic flow (Dury
et al. 1999). Therefore this model was expected to underestimate the pneumatic
conductivity. The van Genuchten-Mualem model (VGM), with the concept of cut
and rejoined pores reflected the effects of geometry better since the combination
of different pore radii in series result in an effective conductivity depending on the
ratios of radii and lengths. For both models, the nonwetting phase conductivity was
predicted from an effective air saturation which was the pore space complementary
to the effective wetting phase saturation. For the emergence point model (EP)
the air content was rescaled by equation (2.64) to reflect the effective mobile air
saturation. The additional free parameter of the EP model, the emergence point
water saturation, was fitted for the homogeneous materials to measured data. For
heterogeneous samples all values evaluated from the homogeneous materials were
used for simulations.
For an overview of underlying measured data, all resulted data of water dynamic
simulations and modeling of pneumatic properties were plotted together (Figure
3.9). The top figure shows the hydraulic data measured in the multistep outflow experiment as symbols and the simulation results for hydraulic functions based on the
estimated parameters as solid lines. The central part shows the temporal evolution
of water saturation in a vertical profile of the sample. Nodes with saturation higher
than emergence point saturation are edged in black. The lower part of the figure
shows the air conductivity as measured during the multistep outflow experiment as
symbols and the simulated values for the specific water saturations with the VGM,
VGB, and EP models, respectively. All plots are connected using the same time
axis.
64
3.6 Calculation of Air Conductivity
5
4
0
-5
-10
bc
outflow
3
2
-15
-20
1
-25
-30
0
0
2
4
6
8
10
t
100
12
10
80
8
sat
h
60
6
40
4
PSfrag replacements
0
0
0
2
4
6
8
10
t
EP
200
VGM
150
VGB
k
ative outflow [cm]
pressure [ cmWC ]
nductivity [cm/h]
time [h]
height [cm]
VGM
VGB
EP
ter saturation [%]
20
2
100
50
0
2
4
6
8
10
t
Figure 3.9: Example of data evaluation of combined multistep outflow and air-flow
measurement. Top: Experimental data from multistep outflow experiment as symbols (dashed line: lower boundary pressure, ◦: outflow,
4: tensiometer) and results of inverse simulation as solid lines. Center:
Temporal evolution of the water saturation of a vertical profile in the
sample which is discretized in 60 nodes. The nodes with a black frame
have a saturation greater than the emergence point saturation. Bottom:
Simulations of the pneumatic conductivity by the VGM, VGB, and EP
models, based on the water saturation shown above. All three plots use
the same time axis which makes a comparison of results possible.
65
4 Experimental Investigations with
Artificial Porous Media
A major hurdle when looking into fluid flow through soil, is to keep the pore space
stable for a prolonged time. In natural soils this is particularly difficult for high
water saturations because clay minerals cause soil swelling and the typically high
water fluxes lead to internal erosion. During drainage, the soil shrinks and the air-
Figure 4.1: Structured porous medium made of sintered glass granules. The sample
consists of two annular layers where the bottom layer is fine textured
and the top layer has an intermediate pore size. The core in the center
is coarse textured. The color is from an experiment with a Brilliant Blue
dye tracer which was transported by capillary rise.
66
4.1 Homogeneous Sintered Glass Columns
flow measurement is obstructed by the by-passing of air at the sample boundaries.
In addition, there is always the problem of microbiological growth and associated
clogging of soil pores in natural soils (Caputo 2000), which can be prevented by the
usage of chemicals such as AgNO3 , mercuric chloride, thymol, and other poisons.
These solutes though, influence the fluid properties and also have to be taken into
account when analyzing the measured data.
Since these processes cannot be suppressed sufficiently in soils, the problems were
circumvented altogether by working with columns made of sintered glass: they were
rigid by construction, the same sample could be examined several times as structure
and texture were constant, and possible precipitates could be removed effectively1 to
have always a homogeneous influence of surface properties (Ustohal et al. 1998). The
smooth surface of the glass beads reduced microscale effects which was in agreement
with experiments carried out by Dullien et al. (1988).
4.1 Homogeneous Sintered Glass Columns
The investigated columns were made of sintered boron silicate glass by ROBU Glasfiltergeräte GmbH, Germany 2 . The chemical composition and some physical properties can be found in Appendix B. The sample size of the cylindrical columns was
limited to a maximum height of 120 mm, and a diameter of 50 - 55 mm for technical
reasons as during the sintering process all material had to be heated up to melting
temperature to allow sintering in the center, while preventing clotting of the glass
granules at the boundaries.
The manufacturer provides several pore size classes with almost uniform grain size,
which results in a narrow pore size distribution. For construction of heterogeneous
structures glass granules of the P16, P100, and P250 classes were for technical reason
used as they do not mix before sintering3 .
The study of the influence of the structure of a sample required knowledge of the
individual elements building the structure. Although it is possible to get structure
information by x-ray tomography of a sample (Vogel et al. 2002) the opposite way
was chosen for this study and the structure was sampled from materials with known
hydraulic properties. For this purpose, a set of homogeneous columns were prepared
and their specific hydraulic properties were examined. Technical data of the columns
is shown in Table 4.1.
The porosity measurement was done gravimetrically. The samples were first oven1
After a longer period of no measurement, the saturated conductivity of the columns was lower
than measured for the new columns. After cleaning the samples with acetone and hydrochloric
acid the original conductivity was measured again.
2
www.robu.com
3
The nomenclature is provided by ISO 4793 and refers to the nominal pore diameter.
4
provided by the manufacturer
67
4 Experimental Investigations with Artificial Porous Media
sample (dimensions)
P16 (h=8.9 cm, ®=4.5 cm)
P100 (h=9.8 cm, ®=5.5 cm)
P250 (h=9.6 cm, ®=5.5 cm)
porosity [%]
48±8
43±2
44±4
Ksat [cm/h] pore diameters [µm]4
0.44
8 - 16
1.8
40 - 100
7.2
160 - 250
Table 4.1: Material properties of the sintered glass columns. Methods used for analyzing are described in text.
dried for 24 hours5 at a temperature of 150 ‰, weighed and afterwards saturated in
a desiccator for 24 hours at a pressure of −500 hPa to ensure no residual air in
the columns6 . The errors in porosity values were mainly caused by the volume
measurement as the columns had no ideal cylindric shape.
Saturated conductivity, Ksat , was measured with the falling head method (Section 3.3). The samples saturated in the desiccator were sealed water- and air-tight
at the sides of the cylinders while the bases left open. Additional sealing in the
permeameter prohibited any by-passing at the boundaries.
While the porosity was almost the same for the three materials, the hydraulic
conductivity increased from P16 to the P250 material due to the increasing pore
size.
As pore scale processes were analyzed in a companion study (Schulz 2003), pore
scale topology of the glass media was examined by scanning a small subsample of
1 cm3 by X-ray tomography with a resolution of (15 µm)3 per voxel. Results for
the P100 and P250 media are shown in Figure 4.2. The scanned materials were
homogeneous and the measured pore radii met the specifications provided by the
manufacturer. The pore structure apparently consisted of pore bodies (large black
areas in Figure 4.2) connected by smaller necks. This might have had a strong
impact on the pressure saturation relation and especially on hysteresis phenomena
as discussed in Section 2.2.4. A closer investigation of the pore geometry of the
sintered glass material and its effect on hydraulic properties can be found in Schulz
(2003).
A tomographic scan of the whole columns showed an internal porosity gradient
in the P250 column, which was due to the sintering technique. At the upper and
lower ends of the column porosity was 0.37 where in the center part a value of 0.48
was measured. This effect did not occur in the other columns.
With x-ray tomography it was not possible to scan the texture of the P16 material. This material was fully saturated in all experiments carried out and therefore
explicit information about pore geometry or unsaturated hydraulic properties were
not relevant for this class.
5
6
Drying for another 72 hours had led to no further reduction of weight.
For extraction of air it was better to increase the applied pressure difference than to keep it
longer in the desiccator.
68
4.1 Homogeneous Sintered Glass Columns
PSfrag replacements
1 mm
L1
L1
Figure 4.2: X-ray tomograms of the sintered glass media in grey-scale picture (solid:
white, void: black). Left: P250 pore size class with pore diameters
of 160-250 µm. Right: P100 pore size class with pore diameters of 40100 µm.
The multistep outflow experiments and evaluation of hydraulic properties which
were carried out for two of the homogeneous media, P100 and P250 are presented in
the following section. An averaging approach for a composition of a heterogeneous
sample consisting of the three materials is shown in Section 4.2. For the heterogeneous sample the simultaneous airflow measurement was also applied.
4.1.1 Homogeneous Glass Column – P250
The multistep outflow experiment for the determination of hydraulic properties of
the P250 glass medium was repeated several times during the experimental stage
of the setup. Since this experiments were done with different boundary conditions
they could be optimized for this sample to improve parameter evaluation.
The column which was wrapped in a PVC tube at the side boundaries was saturated in a desiccator for every single measurement. With the aim of measuring
primary drainage curves, they all had to start at full water saturation. As the column drained freely, it was additionally wrapped in a latex casing for the transport
from the desiccator to the sample holder. After installing the sample in the water
filled sample holder the latex case was removed. Although this procedure was time
consuming (and sometimes quite wet) air inclusions in the installed sample were
reduced to a minimum. The effect of entrapped air became visible when hydraulic
conductivity was reduced noticeably.
69
4 Experimental Investigations with Artificial Porous Media
The lower boundary pressure scheme which was applied to this sample is shown
in Figure 4.3 as a blue dashed line. Due to the large pores and the small width of
the pore size distribution the steps started at 8 cmWC with increments of −2 cmWC
for each step. Several measurements with this column showed that the sample
was almost drained at pressures less than −10 cmWC , therefore imbibition started
at a lower boundary pressure of −12 cmWC after a four hour redistribution phase.
Duration of imbibition steps was set to eight hours for redistribution in the sample
because of the low hydraulic conductivity in the dry range. With this, duration
equilibrium was almost reached for the single steps.
The tensiometer developed for the rigid media (Section 3.2.2) was placed at the
center of the top of the column. Improvement of hydraulic contact was achieved by
use of a layer of gaze and a chuck which fixed the instrument.
Reproducibility of measurements was examined when the experiment was repeated
several times. With the described sample preparation procedure initial saturation
was assumed constant for all experiments. Experiments with different boundary
conditions led to comparable results in the hydraulic functions. For larger steps,
compared to the pore size distribution of the medium, the estimations of hydraulic
parameters were comparable between separate experiments, a result which was already reported (e.g. Hollenbeck and Jensen 1999). Additionally the measurement
was done in a separate setup where only multistep outflow experiments could be
carried out. The results of this measurement corresponded to the measurements in
the new setup.
Experimental Results
The results of the multistep outflow measurement carried out with the P250 sample
at the new setup are shown in Figure 4.3. The outflow curve (◦) showed an air
entry point at a lower boundary pressure of 2 cmWC . Subsequent steps had almost
the same height, as the water table was drawn through the column until the lower
boundary pressure was at −6 cmWC . At −12 cmWC the sample was drained to a
residual water content of 12 %. Further reduction of pressure gained no significant
increase of water content as measured in separate experiments.
The rewetting of the sample started after a four hour redistribution phase. Since
the hydraulic conductivity was low at this saturation, there was no equilibrium
within the sample. Therefore the first imbibition steps showed a further increase of
outflow until the wetting direction had changed from drainage to imbibition throughout the whole column. Imbibition showed significant hysteresis in the pressure saturation relation, since the water content was almost at its minimum value until a
lower boundary pressure of −4 cmWC was applied. The sample was imbibed in four
steps of almost equal heights which can be interpreted as the advancing wetting
front in the column.
After measurement of a complete out- and inflow cycle an air content of 5 %
enclosed in the column was measured by weighing the sample before and after re-
70
4.1 Homogeneous Sintered Glass Columns
saturation in a desiccator.
Hysteresis effects could be explained by the geometry of the pore space (Figure
4.2, left), which apparently corresponded to larger pore bodies connected through
narrow necks. The latter, were critical during drainage while the former, were critical
during imbibition (Figure 2.7).
Hydraulic Properties
For the evaluation of hydraulic properties the eshpim fitting code was preconditioned
with all data available from separate experiments. For the rigid medium saturated
water content was set to measured porosity. Measured saturated conductivity was
set as an initial value and the fitting routine was allowed to adjust this parameter
within one order of magnitude up and down. For αd , αw , and n, the initial values
were evaluated from a static pressure saturation measurement and the initial value of
parameter τ was set to 0.5. The hysteresis effect was considered by the constrained
Kool and Parker model. Results of the inverse parameter estimation are listed
in Table 4.2 together with their respective confidence intervals. The calculated
4
10
5
3
-5
2
L4
L2
0
-10
PSfrag replacements
sure [ cmWC ]
outflow [cm]
time [h]
1
-15
-20
0
0
20
40
60
L1
80
100
120
Figure 4.3: Measurement and simulations of hysteretic outflow curve of P250 glass
column. Lower boundary pressure (dashed), cumulative outflow (black),
and tensiometer potential (green), as measured (symbols) and fitted
(solid lines).
71
4 Experimental Investigations with Artificial Porous Media
covariance matrix did not show any significant correlation between any parameters
for this fit.
The simulated outflow curve is shown in Figure 4.3 (solid black line). Irregularities
between simulation and measured data in the pressure ranges where the wetting front
moves through the column, reflect the porosity gradient in the column which was
measured by x-ray tomography. Since simulation had to be done with homogeneous
porosity this variation could not be taken into account. Discrepancies between
measured tensiometer data and the simulated potential at the upper end of the
column was also a result of the porosity gradient within the P250 sample. Since
the difference in porosity was a result of the sintering procedure, the pore size
distribution changed at the upper end towards smaller pores. As a result, hydraulic
conductivity was higher in this part of the sample than estimated by the model
which was based on a homogeneous porosity.
The water dynamics at the reversal point was reflected well by the simulation.
The outflow curve was decreasing first, because the lower part of the sample was
imbibed and then the curve was rising again when that part of the sample was
drained where pressure was still lower than the actual pressure.
The resulting hydraulic functions for the P250 column are shown in Figure 4.4
(green lines). The pressure saturation relation (left) and the pressure conductivity
relation (right) were calculated with the parameters estimated from the transient
experiment for the drainage and imbibition branches of the multistep outflow curve.
The steep slope of the pressure saturation relation reflects the narrow pore size
distribution of the sintered glass material. Note that the material properties shown
in the figure do not reflect the sample height (as they should be independent), i.e.
the hydrostatic pressure assigned to the sample height has to be added before the
results are compared to measured data.
sample
P250
parameter
αd [cm−1 ]
0.429±0.011
n
3.63±0.14
8.8±0.2
Ksat [cm/h]
θs
0.44
0.01±0.01
θr
τ
0.61±0.14
0.142±0.002
αw [cm−1 ]
Table 4.2: Van Genuchten parameters for homogeneous P250 glass sample as estimated with eshpim and their respective 95 % confidence interval. The
value of θs was fixed for simulations to the measured value.
72
PSfrag replacements
θ
uctivity [cm/h]
logψ[ cmWC ]
0.45
9
0.40
8
0.35
7
0.30
6
5
0.25
k
theta
4.1 Homogeneous Sintered Glass Columns
0.20
0.15
0.10
0.05
0
PSfrag replacements
θ
conductivity [cm/h]
logψ[ cmWC ]
-1
1
0
2
4
3
2
1
0
-1
psi
1
0
2
psi
Figure 4.4: Hysteretic hydraulic functions of two homogeneous glass columns P250
(green) and P100 (red) estimated with the constrained Kool and Parker
model, for both drainage (solid curve) and imbibition (dashed) branches.
Separate fit of imbibition branch of the P100 glass (blue). Left: Pressure
saturation curve. Right: Pressure conductivity curve. The shaded areas
show the range of measurement for the P250 sand (light grey) and P100
(additional dark grey).
4.1.2 Homogeneous Glass Column – P100
The preparation of the P100 sample was almost the same as for the P250 column.
The sample was wrapped in a PVC tube and saturated in a desiccator. For this
column the problem of free drainage during installation did not occur due to its
smaller pores. Lower boundary conditions for the measurement are shown in Figure 4.5. Starting at 1 cmWC , drainage was carried out to a maximum pressure of
−49 cmWC with a variable duration of steps due to the changing hydraulic conductivity. Although the inverse parameter estimation did not require steady states, the
uncertainty of the parameters was decreased when the shape of the outflow curve
contained both parts of the system response, the fast rise in the beginning of a new
pressure step and the slow redistribution. For imbibition, the heights of the pressure
steps were the same, and duration was chosen with respect to the changing hydraulic
conductivity.
The external tensiometer was placed in the center on top of the column and fixed
with the chuck.
Experimental Results
The measured outflow curve and tensiometer pressure data is shown in Figure 4.5
(symbols). Drainage started at a water potential of −10 cmWC which corresponds
to the theoretical air entry pressure based on the pore size which was measured in
the tomograph. The single outflow steps did not reach equilibrium like in the P250
73
4 Experimental Investigations with Artificial Porous Media
4
Sfrag replacements
L2
-20
2
-30
1
-40
0
0
10
20
30
40
L1
50
60
70
80
4
Sfrag replacements
0
-10
3
L2
-20
2
L3
outflow [cm]
ure [ cmWC ]
time [h]
ure [ cmWC ]
outflow [cm]
time [h]
-10
3
L3
outflow [cm]
ure [ cmWC ]
time [h]
ure [ cmWC ]
outflow [cm]
time [h]
0
-30
1
-40
0
0
10
20
30
40
L1
50
60
70
80
Figure 4.5: Measurement and simulations of the hysteretic outflow curve of the P100
glass column. Lower boundary pressure (dashed), cumulative outflow
(◦), and tensiometer potential (4), as measured (symbols) and fitted
(solid lines). Simulations of the primary drainage and imbibition curves
with constrained Kool and Parker hysteresis approach (top) and separate
sets of parameters for each branch of the curve (bottom).
material. At a pressure of −49 cmWC drainage was stopped. After a 4 hour redistribution phase, residual water in the column was 7 %7 . Due to the low hydraulic
conductivity in the dry range the reversal from drainage to imbibition developed
very slowly in the column. The cumulative outflow was increasing slightly for another ten hours and the tensiometer pressure was also not responding to the higher
pressures. When water was flowing back into the column the tensiometer pressure
showed a jump when hydraulic conductivity increased again. The retention relation showed strong hysteresis as the minimum water content was kept to a lower
boundary pressure of −24 cmWC . The main part of resaturation was done during
7
Since the experiments were repeated several times, it was possible to interrupt the cycles and
measure water content between drainage and imbibition.
74
4.1 Homogeneous Sintered Glass Columns
the last two pressure steps. At the end of the experiment a residual air content of
4 % was measured gravimetrically before and after resaturation of the column in a
desiccator. Possibly this amount of air could have been further decreased when the
last step would have been lasted longer.
Hydraulic Properties
Parameter estimation for this measurement was done with two different approaches.
The fitting results are shown in Figure 4.5 (top) for the constrained Kool and Parker
model and (bottom) for the Kool and Parker model with a full set of parameters
for both branches.
Evaluation of the measurement was first done with the constrained model which
was provided by eshpim. To achieve the best agreement of the fit and measured
data, the outflow curve was fitted solely and the parameters were fixed. Except of
the five hours in the beginning of the experiment where the system did not meet the
requirements for the Richards equation this fit described measured data well. The
entire cycle was then fitted with the the second α as free parameter for imbibition.
The constrained model failed in description of the system during rewetting. Neither
the outflow nor the tensiometer data could be fitted by this approach. The results
for this parameter estimation are listed in Table 4.3 and the simulations based on
this set of parameters are shown in Figure 4.5 (top).
To get more flexibility in the hydraulic functions for both branches of the multistep outflow experiment the fitting of data was split into two separate parts. The
outflow branch was fitted with a complete set of parameters to an optimum agreement. Estimation of hydraulic properties was done with a full set of parameters
for the imbibition branch of the multistep outflow curve. Simulation results for this
Kool and Parker (1987)
parameter
αd [cm−1 ]
n
Ksat [cm/h]
θs
θr
τ
αw
0.033
5.18
2.6
0.43
0.00∗
0.89
0.097±0.001
separate branches
drainage
imbibition
0.033±0.001
5.18±0.04
4.03±0.09
2.6±0.1
2.6
0.43
0.43
0.00 ∗
0.00 ∗
0.89±0.05
−1.96±0.28
0.129±0.001
Table 4.3: Van Genuchten parameters for homogeneous P100 glass sample as estimated with eshpim and their respective 95 % confidence intervals. Hysteresis was considered by the constrained Kool and Parker model and by
separate simulation of each branch of the measured curve. The value of
θs was fixed for simulations. ∗ Parameter reached limit.
75
4 Experimental Investigations with Artificial Porous Media
technique are presented in Figure 4.5 (bottom). The thin grey line separates both
branches used for fitting. With this method the slow redistribution after reversal
from drainage to imbibition was simulated well for the outflow and the tensiometer. The major deviation from measured tensiometer data was the prediction of the
jump in the hydraulic conductivity at the tensiometer position after 61 hours which
was measured two hours later in the experiment when the next pressure step was
started. This deviation could be the result a boundary effect in simulations since
the tensiometer was placed at the top of the sample. The outflow curve was also
better fitted by this model due to the additional degrees of freedom. The residual
air content was slightly overestimated which could be reduced for an experiment
which ends at zero potential at the upper boundary to reach starting conditions.
Parameters which resulted from this fit are listed in Table 4.3 (right).
Figure 4.4 shows the hydraulic functions for the P100 material. The constrained
Kool and Parker model is represented by the red lines. The imbibition curves (red,
dashed) are parallel to drainage (red, solid) as they were based on the same van
Genuchten n parameter. The steep slope of the retention function reflects the narrow
pore size distribution of the material which was measured by x-ray tomography. The
result of the separate simulation of both branches is the additional imbibition curve
(blue, dashed) which has a slope different from the corresponding drainage curve
(red, solid). The difference in the slope of this curves, i.e. the different nd and ni
parameters, could take the different behavior in the drainage and imbibition process
into consideration (Section 2.2.6.3). A problem which is suppressed by the constraint
is the crossing of the curves which could not be explained on physical principles for
the main drainage and imbibition branches (Kool and Parker 1987, Schultze 1998),
but this effect did not occur with this material.
4.2 Structured Column
For examination of column scale heterogeneity effects, a fourth column was composed made of the three homogeneous materials, introduced in the previous section.
The major part consisted of P100 material where two lenses were embedded. The
P250 and P16 materials were used for building the lenses (Figure 4.6, Table 4.1). The
lenses were 5 mm flat slices with the same diameter as the whole column. To make a
permanent flow in the P100 material possible, a part of the slice was cut-off as shown
in the figure. The volume of each lens was 8 % of the sample volume which made
the effects of each lens visible without dominating the processes in the column. One
reason for this kind of structure was its effect on residual fluid saturations, where
the volume had to be large enough to distinguish between structure and texture
effects. The lense should offer a compartment for residual water during drainage
and the coarse lense should contain the residual air in the imbibition branch of the
experiments. For this purpose the column was flattened at both ends and could
76
4.2 Structured Column
L1
L2
L3
L4
PSfrag replacements
40 mm
120 mm
83 mm
78 mm
40 mm
35 mm
49 mm
L5
L6
L7
Figure 4.6: Structured sintered glass column. The sample consists of the three materials: P100 (light grey) with two lenses of P250 (black) and P15(white),
which are 8 % of the sample volume each.
have been installed with the coarse lens up and with the coarse lens down. Another
aspect referred to different flow regimes caused by varying saturations which has an
enormous impact on solute transport processes. A comparison of two characteristic
cases is presented in Appendix B.3.
For simultaneous measurement of hydraulic and pneumatic properties, the sample
was installed with the coarse lens up. The applied lower boundary pressures were
similar to the pressures used for the P100 experiment but steps were varied in length
as the local equilibrium states were not required. The steps in the beginning of
the experiment where the hydraulic conductivity was high lasted for one hour only,
whereas the length of the steps was increased in the drier range. The duration of the
experiment was optimized with the prior knowledge of the system response provided
by the P100 experiment. For resaturation where a low hydraulic conductivity was
expected in the beginning the first imbibition steps lasted only two hours but the
77
4 Experimental Investigations with Artificial Porous Media
10
4
18
5
16
0
14
3
-5
12
L2
-15
rag replacements
e [ cmWC ]
utflow [cm]
time [h]
ity [cm/h]
-20
1
0
0
10
20
30
40
50
60
70
10
L3
2
L4
-10
8
6
-25
4
-30
2
-35
0
80
L1
Figure 4.7: Measurement and simulations of hysteretic outflow curve of structured
glass column. Lower boundary pressure (dashed), cumulative outflow
(◦) tensiometer potential (4), and air conductivity (red). Simulation
results for outflow and tensiometer curves represented as solid lines in
the respective color. The pressure jump of the simulated tensiometer
pressure was due to the inconsistency in the parameterization for the
two branches.
later steps were held for eight to ten hours. Air-flow measurement was carried out
when lower boundary pressure in the water phase was equal to and less than 0 cmWC .
The whole pressure scheme applied to this sample is shown in Figure 4.7 as blue
dashed line.
Experimental Results
The experimental results for the combined multistep outflow and air-flow measurement are shown in Figure 4.7. The air entry was at a lower boundary pressure
of −10 cmWC as expected for the P100 material. As a major difference from the
P100 material the hydraulic conductivity was lower for this sample, which could be
attributed to the hydraulic conductivity of the P16 lens (Table 4.1). The system responded very slowly to the applied pressure changes. The tensiometer did not follow
the lower boundary pressures when the −25 cmWC step was applied after 10 hours.
Although the steps lasted for 10 hours steady states were not nearly reached. After
the reversal to imbibition water was flowing back into the column immediately. The
retention relation showed hysteresis between the drainage and imbibition branches.
78
4.2 Structured Column
At a lower boundary pressure of 10 cmWC in the last imbibition step, there was
about 9 % of residual air in the sample. This amount of air was assumed to be
entrapped in the coarse lense during imbibition. Consequently, hysteresis was observed at two different scales: structural hysteresis as a result of the macroscopic
lenses, and textural hysteresis due to the geometry of the pores of the different materials. The air was then removed slightly by continuous air paths from the lens
to the atmosphere. The experiment was stopped too early to get the final residual
air content. For the measured state the residual air saturation was 6 % which was
measured gravimetrically before and after saturation in the desiccator.
At a pressure of −20 cmWC air became continuous and the air-flow measurement
started. The corresponding water content was 20 %. Air conductivity increased
during drainage following the stepwise reduction of water content in the column.
At a water content of 8 % the pores which were relevant for air-flow were drained
and a further reduction of water content resulted only in a slight increase of air
conductivity. The pore water which was drained in the following steps came from
small tortuous pores which had little impact on the integral air permeability. In
the imbibition branch the maximum air conductivity was held. Extinction of air
conductivity could not be measured in this setting as the respective saturation was
reached at water potential higher than zero where air conductivity could not be
measured anymore.
Hydraulic Properties
For the type of heterogeneity investigated in this experiment it was evident that
the structure of the sample played a specific role for the effective hydraulic and
pneumatic properties at a larger scale and that simple averaging of properties of
homogeneous materials could not be possible.
The simulation of water dynamics of the structured column was based on the prior
knowledge of the hydraulic properties of the homogeneous materials. The simulation
results presented in Figure 4.7 were a result of a separate fitting of both branches.
Since the applied pressure was too low for drainage of the P16 lense, the bimodal
van Genuchten model was used for simulations. The material properties, i.e. the α’s
and n’s of the homogeneous materials P100 and P250 were fixed and initial values
of the other parameters were set as they were measured in separate experiments.
The parameters resulted from the best fit are listed in Table 4.4.
Because of the horizontal structure of the sample, it did not offer best conditions
for a bimodal fit. The simulations with a fixed weight of the two materials resulted in
an early outflow which was not observed. In the outflow branch the coarse textured
material was assumed to be in contact with the air reservoir and therefore drained
when the air entry pressure of this material was reached. This problem was reduced
when the weight of the two modes was also fitted. The best fitting results were
achieved with a weight of 3 % for the volume of the coarse textured material. The
saturated hydraulic conductivity, which was also fitted, reached a value comparable
79
4 Experimental Investigations with Artificial Porous Media
parameter
α1 [cm−1 ]
n1
α2 [cm−1 ]
n2
wi
Ksat [cm/h]
θs
θr
τ
drainage
imbibition
0.033
0.129
5.18
4.03
0.142
0.429
3.63
3.63
0.97±0.01
80∗
0.33±0.01
0.33
0.43
0.43
∗
0.00
0.00∗
−0.14±0.06 −1.61±0.13
Table 4.4: Bimodal van Genuchten parameters for heterogeneous glass sample. Material properties of the homogeneous materials were used to simulate heterogeneous structure. Free parameters as estimated with eshpim and
their respective 95 % confidence intervals. ∗ Parameter reached limit.
to that measured for the P16 material. This was also a result of the horizontal
structure where the fine textured lens delimited the conductivity of the sample.
With this set of estimated parameters the outflow curve and the tensiometer curve
could be fitted with a good agreement.
For simulation of this experiment, the approach of separate fitting of both branches
in combination with the bimodal model failed for the imbibition branch. On the
one hand, there was the influence of the non-steady state at the reversal point. The
change of parameters was done for the whole column and that part of the sample
which was still in the drainage process would have required the respective parameters
until the reversal was carried out. This led to errors in the saturation distribution in
the beginning of the simulations, which were affecting the whole imbibition branch.
On the other hand, the horizontal sample structure also had a crucial impact on the
simulations of the rewetting branch. Because there was no direct connection to the
air reservoir, there was some entrapped air in the coarse lens which was removed
very slowly. This type of back emptying was also observed by Mortensen et al.
(2001), who explained the slow processes by a flow through single channels, which
could not be wetted since the airflow blocked the water. The fitting of the curve
resulted in a weight for the coarse material which increased to the upper limit of
20 %. This value could not be related to any measured data and therefore seems
to be useless. The estimated parameters are listed in Table 4.4. The hydraulic
functions shown in Figure 4.8 show only the drainage branch of the simulations.
The bimodal hydraulic functions reflect the two maxima in pore size distribution,
as assumed by the model. Compared to the homogeneous P100 column, the impact
of the coarse P250 material on the bimodal conductivity function was a decrease in
the range from zero to −15 cmWC by 25 %.
80
PSfrag replacements
PSfrag replacements
PSfrag replacements
nductivity
relative
[-] conductivity
relative
[-] conductivity [-]
S [-]
S [-]
S [-]
g ψ [ cmWC ] log ψ [ cmWC ] log ψ [ cmWC ]
Sw [-]
Sw [-]
Sw [-]
1.0
1.0
G
1.0
G
0.9
0.7
0.7
0.6
0.6
0.6
F
0.4
0.3
0.2
0.1
0
0
2
1
psi
vG
VGM
VGB
EP
0.5
P
0.3
D
0.2
3
F
0.4
0.1
0
0
2
1
psi
0.8
k
0.8
0.7
0.5
G
0.9
0.8
k
S
0.9
vG
VGM
VGB
EP
4.2 Structured Column
0.5
0.4
F
P
0.3
P
D
0.2
D
vG
VGM
VGB
EP
3
0.1
0
1
0
theta
Figure 4.8: Bimodal hydraulic and pneumatic functions for outflow branch of the
heterogeneous glass column. Pressure saturation relation for water and
air (left) and corresponding pressure conductivity (center) and saturation conductivity relations (right).
Pneumatic Properties
Based on the simulations of the water dynamics for the drainage branch of this
experiment, the pneumatic properties were evaluated like introduced in Section 3.6.
The temporal evolution of a saturation profile was calculated assuming homogeneous distribution of the two materials involved in drainage. The sample height was
resolved in 73 nodes, where the lower six represented the separator plate. Simulated
water saturations reflecting the multistep outflow experiment are shown in Figure
4.9 (center) for the nodes within the sample.
Air conductivity was calculated for the discrete nodes of the simulation according
to the VGB, VGM, and EP models. For the simulations only hydraulic parameters
were used. A comparison of the simulated and measured conductivities is shown in
the lower part of Figure 4.9. The VGB and VGM models did not fit measured data
very well. Both models assumed that effective air saturation was the pore space
complementary to the water saturation (Figure 4.8, left). Beginning with drainage,
a nonzero air conductivity was calculated for this models, which was not observed
in experimental data. The difference in the rise of air conductivity reflected the
different terms for pore size and tortuosity in the conductivity relations (Section
2.3.3). In the VGM model tortuosity is better reflected but the transition from
hydraulic to pneumatic properties was still not possible. For the EP model airflow through the sample did not start until the emergence point water content of
θm,ep = 12 % was reached in the single nodes of the simulation. This value was
fitted in the simulations to achieve best agreement to measured data. Nodes, where
saturation was higher than this emergence point saturation, are marked with a black
frame in Figure 4.9 (center). As measured data showed the effect of a spontaneous
starting air conductivity, this type of model was most suitable. The stepwise rise of
81
4 Experimental Investigations with Artificial Porous Media
5
0
4
-5
-10
-15
bc
outflow
3
-20
2
-25
-30
1
-35
-40
0
0
5
10
15
20
25
30
35
40
t
100
12
10
80
8
sat
h
60
6
40
4
20
2
g replacements
0
5
10
15
20
25
30
35
40
t
20
15
k
flow [cm]
[ cmWC ]
ty [cm/h]
time [h]
ight [cm]
0
0
VGM
VGB
EP
10
EP
VGM
5
VGB
ation [%]
0
0
5
10
15
20
25
30
35
40
t
Figure 4.9: Simulation of air conductivity in structured glass column. Top: Measurement and simulations of hydraulics (outflow branch). Center: Temporal
development of saturation in a vertical profile of the sample, based on
the P100 material. Bottom: Air conductivity calculated for three air
conductivity models. Nodes with saturation more than emergence point
saturation are framed black.
82
4.3 Discussion
air conductivity was predicted well by the model, until the measured data reached its
maximum after 21 hours. The overestimation of maximum air conductivity could be
attributed to an effect caused by the structure of the sample. The one-dimensional
model did not account for the fine P16 lens, which kept saturated during the entire
experiment. This lense reduced the cross sectional area which was available for airflow. Continuous air paths could be established only in the P100 material with the
small cross sectional area within this layer of the sample.
Figure 4.8 shows the comparison of the hydraulic and pneumatic functions for
the used models. The VGM and VGB models were based on an air saturation
complementary to water saturation, whereas the EP model showed a sharp rise in
the retention function when the emergence point saturation was reached (Figure 4.8,
left). In the comparison of the conductivity models with measured data the steep
slope of the data could not be predicted by any model in the pressure conductivity
relation (center). This could be caused by the discrete water pressure steps which
were applied. The saturation conductivity relation was also predicted best by the EP
model (right). Both the VGB and VGM models underestimated the air conductivity
over a wide range of saturation. The tortuosity exponent gave these curves their
specific shape which is different from the shape of the EP model prediction with the
tortuosity factor estimated form hydraulic data.
4.3 Discussion
The experiments with homogeneous and heterogeneous sintered glass columns have
shown that the multistep outflow technique could be combined with an additional
instrument for the determination of pneumatic properties. The measurement of homogeneous samples with the combined setup, resulted in hydraulic properties which
were the same as measured in a parallel setup for multistep outflow measurements
only.
The influence of structure became visible in the heterogeneous sample. Although
the bimodal model should reflect the materials involved in the experiment, it was
not possible to predict the whole drainage imbibition cycle. The superposition of the
homogeneous hydraulic functions required a direct connection of all compartments
to the air and water reservoirs which was not given due to the horizontally layered
structure. This seemed to be strongly required especially for the sintered glass
materials with the narrow pore size distributions.
As it was possible to simulate the water dynamics for the drainage branch of the
multistep outflow experiment, it was possible to simulate the air conductivity with
the VGM, VGB, and EP models. A comparison of measured data and simulation
results showed a better agreement with the EP model. A rescaling of the effective
saturation seemed to be mandatory for the predictions of air conductivity since the
simple assumption of an effective air saturation which is complementary to the water
83
4 Experimental Investigations with Artificial Porous Media
saturation was insufficient.
The rigid sintered glass media had a major advantage during the stage of the
development of the experimental device. Experiments were repeated several times
always with the same sample. Deviations of experimental results which could not
be explained in a physical context, were detected as measurement artifacts and led
to improvement of the setup. On the other hand, the rigid material had a major
disadvantage at the interface between sample and separator plate. The problem of
gaps which made hydraulic contact worse demanded an extremely careful installation
of the samples in the sample holder.
84
5 Two Phase Flow Measurement
with repacked Sand Columns
As a second class of materials, sieved repacked sands were examined. With the
aim of investigating several structures in the sand, sets of homogenous sands were
analyzed in homogeneous packings. With the known properties of the homogeneous
materials structured repacked columns were for experimental examination and numerical simulations realized to focus on specific effects caused by the texture of the
components and the structure of the samples.
5.1 Characterization of Homogeneous Sands
The base materials for the sand packings were of coarse and fine sand referred to as
S-I and S-II, respectively. The sand, sieved and washed, was at spherical shape, as
it had originally come from a river. The two grain size classes of the sands and the
dimensions of the homogeneous samples are listed in Table 5.1.
Assuming the homogeneity of the sand of separate samples, saturated conductivity
was measured with the falling head method in an extra sample of the respective
material. Porosity also was calculated via bulk density measurement in separate
samples.
sample (dimensions)
porosity [%]
S I (h=10.0 cm, ®=18.0 cm)
32±5
S II (h=6.3 cm, ®=16.8 cm)
38±5
Ksat [cm/h]
200.0
25.0
grain size [µm]
630 - 1250
250 - 630
Table 5.1: Material properties of the homogeneous sand columns. Methods used for
analysis are described in text.
5.1.1 Coarse Sand – S-I
Hydraulic properties of the coarse sand S-I were determined in a 28 hour multistep
outflow experiment. The lower boundary pressure was selected with respect to the
85
5 Two Phase Flow Measurement with repacked Sand Columns
large pores. The initially saturated column was drained in seven steps with pressures
of 0.0, −2.5, −5.0, −7.5, −10.0, −15.0, −20.0 cmWC with the aim of getting almost
identical amounts of water in the outflow of each step. The duration of each single
step was one hour except the −5.0 cmWC step which lasted for two hours. After a one
hour redistribution phase at the end of drainage, the column was resaturated. The
duration of each step was increased to 3.3 hours. The applied imbibition pressure
steps were identical to the drainage scheme with two additional steps at the end of
saturation where pressure was increased to +5.0 cmWC and +10.0 cmWC steps for
one hour each, to reach starting conditions.
An internal tensiometer was placed at a height of 8 cm above the lower boundary.
This instrument was a 5 cm long ceramic tube, which was in contact with the sample
over its whole length, and was not of the type introduced in Section 3.2.2.
Experimental Results
The resulting outflow curve is shown in Figure 5.1. The air entry pressure of this
material was reached with the first pressure step. The small step in the outflow
curve at 0.5 hours was due to a small irregularity at the lower boundary pressure.
The magnetic valve which controlled the pressure had been closed too late and the
10
5
2
0
bc
out
-5
-10
1
-15
rag replacements
utflow [cm]
time [h]
re [ cmWC a]
-20
0
0
5
10
15
20
25
t
Figure 5.1: Measurement and simulations of hysteretic outflow curve of sand S-I.
Lower boundary pressure (dashed), cumulative outflow (black), and tensiometer potential (green), as measured (symbols) and fitted (solid lines).
86
5.1 Characterization of Homogeneous Sands
pressure was −1 cmWC for 30 seconds which was enough to drain a specific amount
of water from the sample. Due to hysteresis, the water did not flow back to the
column when the pressure returned to 0 cmWC again. The sample had been almost
desaturated after the last drainage step. The imbibition branch of the curve showed
significant hysteresis in the retention relation since saturation at the −10.0 cmWC
step was almost the same as at the −15.0 cmWC step of drainage. At the pressure of
+10.0 cmWC the column was completely saturated with water, i.e. there had been
no residual air in the sample.
Hydraulic Properties
Figure 5.1 also shows the results of simulated curves for the outflow and tensiometer potential, fitted with the constrained Kool and Parker model implemented in
eshpim. All information about material parameters which were measured separately
was used for preconditioning the parameter estimation. The saturated water content had been fixed to the measured value and the initial value used for saturated
conductivity was that measured with the falling head method while the fitting routine was allowed to adjust this parameter within one order of magnitude up and
down. Initial values of the other parameters were varied for several simulations
within physically usefully ranges. Estimated parameters are listed in Table 5.2 together with their 95 % confidence interval. The parameters were estimated with
different starting conditions to reduce the possibility of hitting a local minimum for
the parameter optimization. All simulations led to results which were similar within
their confidence intervals and a global minimum for parameter identification was
assumed. Statistical analysis resulted in no significant correlation of parameters.
The model was not able to predict the cumulative outflow of the first two drainage
steps. Especially in the wet range and for the homogeneous material with the narrow
sample
parameter
αd [cm−1 ]
n
Ksat [cm/h]
θs
θr
τ
αw
S-I
0.105±0.001
6.76±0.09
156.1±9.0
0.28
0.011±0.001
0.998±0.052
0.078±0.001
Table 5.2: Van Genuchten parameters for homogeneous S-I sand sample as estimated
with eshpim and their respective 95 % confidence interval. The value of
θs was fixed for simulations to the measured value.
87
5 Two Phase Flow Measurement with repacked Sand Columns
pore size range the van Genuchten model could not reproduce the sharp air entry
with the resulting abrupt rise of the cumulative outflow curve (Vogel et al. 2001).
Although the tensiometer simulations fitted the measured data well, measured outflow was less than the model prediction. Beginning with the third step measured
data was fitted well by the model. At maximum pressure, the model predicts outflow which is less than measured. This was also a result of model errors in the dry
range.
During imbibition, dynamic effects were observed beginning with the −8 cmWC
step to the end of the experiment. When a new pressure step had begun there
was a fast decrease of the outflow curve followed by a slow almost linear period.
This linear part was due to reduced conductivity by entrapped air. During the
following redistribution, air was removed from the sample and the dynamics could
be described by the simulations which fitted the measured data again.
Hydraulic functions for the fitted van Genuchten parameters are shown in Figure
5.2 (green lines) for both branches of the hysteretic curve.
16
0.35
14
0.30
12
Sfrag replacements
θ
vity [cm/h]
gψ[ cmWC ]
10
0.20
k
theta
0.25
PSfrag replacements
0.15
0.10
0.05
0
0
θ
conductivity [cm/h]
logψ[ cmWC ]
1
psi
8
6
4
2
0
1
0
psi
Figure 5.2: Hysteretic hydraulic functions of two homogeneous sands S-I (green) and
S-II estimated with the constrained Kool and Parker model (red), for
both drainage (solid curve) and imbibition (dashed) branches. Separate
fit of the imbibition branch of S-II sand (blue). Left: Pressure saturation
curve. Right: Pressure conductivity curve. For better comparison the
hydraulic conductivity of S-I sand is scaled down by a factor of 10 (Ksat =
156.10 cm/h). The shaded areas show the range of measurement for the
S-I sand (light grey) and S-II (additional dark grey).
5.1.2 Fine Sand – S-II
With the homogeneous S-II sand the effects of hysteresis on the hydraulic and simultaneously on the pneumatic properties in a multistep outflow experiment were
88
5.1 Characterization of Homogeneous Sands
analyzed. The measurement was started at full water saturation. The tensiometer
was placed in the middle of the sample 1.5 cm above the lower boundary. Air-flow
measurements were carried out at four positions on each outflow step, the first point
at the outflow phase of the step, the second and third during redistribution and the
fourth point of measurement was carried out before switching to the next pressure
step. After the primary drainage and imbibition cycle two additional scanning cycles were measured to investigate hysteresis effects on the water phase and on the
gaseous phase where the emergence and extinction point saturation could also show
hysteresis effects.
Experimental Results
The lower boundary condition, measured tensiometer potential, cumulative outflow
curve and air conductivity measurements are shown in Figure 5.3. The first cycle had
a duration of 152 hours, followed by the first scanning loop from 152-217 hours and
600
500
400
air
outflow
2
300
1
200
100
0
0
0
50
100
150
200
250
150
200
250
t
5
0
-5
PSfrag replacements
ssure [ cmWC ]
e outflow [cm]
time [h]
ctivity [cm/h]
lb
-10
-15
-20
-25
-30
-35
0
50
100
t
Figure 5.3: Multistep outflow with simultaneous air-flow measurement for sand SII. Top: Cumulative outflow (grey) and air conductivity (red). Bottom:
Predefined lower boundary pressure (blue, dashed) and measured tensiometer potential (green).
89
5 Two Phase Flow Measurement with repacked Sand Columns
a second scanning cycle 217-282 hours. The outflow curve shows pressure saturation
hysteresis for the imbibition branch which had been enlarged during the scanning
cycles. Residual air content at the end of each cycle had also slightly increased due
to the fact that initial pressure was not reached.
For all three cycles there was a clear emergence and extinction of air permeability.
Air-flow began when the gaseous phase was continuous along the sample at a water
content of 16 % which was the integral value for the sample without consideration
of the internal saturation gradient due to the sample height. The macroscopic
extinction point in the primary imbibition curve was measured at a water content
of 28 % measured in a separate experiment which was started at full air saturation.
The scanning cycles reproduced the respective saturation where the emergence point
water content varied between 16-26 % and the extinction point water content was
between 22-30 %.
Because of the vertical arrangement of sample and separator there was always the
effect of a saturation gradient within the column due to gravity. For this reason
the measured saturations had to be seen as effective saturations over the full height
(6.3 cm) of the sample. A detailed analysis of air conductivity will follow later in
this Section.
Hydraulic properties
Estimation of the hydraulic properties of the S-II sand had been made with the same
preconditions used for the homogeneous S-I sand. The saturated water content was
fixed to the measured value and and saturated conductivity was fitted within one
order of magnitude up and down the measured value. Figure 5.4 (top) shows the
simulation results for the primary drainage/imbibition cycle using the constrained
constrained
parameter
αd [cm−1 ]
0.052±0.001
n
5.88
Ksat [cm/h]
15.7
θs
0.38
θr
0.01±0.01
τ
0.79
0.081±0.002
αw
separate branches
drainage
imbibition
0.055±0.001
5.88±0.13
3.01±0.19
15.7±5.2
15.7
0.38
0.38
0.07±0.01
0.03±0.01
0.79±0.18
2.12±0.10
0.082±0.001
Table 5.3: Van Genuchten parameters for homogeneous S-II sample as estimated
with eshpim and their respective 95 % confidence intervals. Hysteresis
was considered by the constrained Kool and Parker model and by separate
simulation of each branch of the measured curve. The value of θs was fixed
for simulations.
90
5.1 Characterization of Homogeneous Sands
5
0
2
-5
-10
L2
PSfrag replacements
ssure [ cmWC ]
e outflow [cm]
time [h]
L3
-15
1
-20
-25
-30
-35
0
0
20
40
60
80
L1
100
120
140
5
0
2
-5
-10
L2
PSfrag replacements
ssure [ cmWC ]
e outflow [cm]
time [h]
L3
-15
1
-20
-25
-30
-35
0
0
20
40
60
80
L1
100
120
140
Figure 5.4: Simulations of the primary drainage and imbibition curves with constrained Kool and Parker hysteresis approach (top) and separate sets of
parameters for each branch of the curve (bottom).
Kool and Parker approach for hysteresis. As it was not possible to fit the entire
curve, a fit of the outflow branch was the basis of the simulation shown in the figure.
Outflow parameters were fixed and the α for imbibition was fitted to maximum
agreement.
For this material, the more flexible functions of both branches were necessary
with respect to the evaluation of the scanning cycles. The result for the separate
evaluation of both branches of the first cycle is shown in Figure 5.4 (bottom). With
this model the experiment could be described well. The outflow curve and the
tensiometer pressures were predicted by the simulations in good agreement with the
measured data. The van Genuchten parameters resulting from this simulations are
listed in Table 5.3. The resulting hydraulic functions are also shown in Figure 5.2
(constrained: red lines, separate fitting: blue lines).
91
5 Two Phase Flow Measurement with repacked Sand Columns
Pneumatic Properties
The measurement of pneumatic conductivity was compared with the three air conductivity models discussed in Section 2.3.4. Figure 5.5 (left) shows the saturations
of the two phases as they were used for the prediction models. The water retention
curve (black) is shown in the figure as to visualize the dependence of air saturation.
The VGM and VGB models were based on the same air saturation curves, where
air occupies the pore space complementary to water. For the EP model, effective
air saturation had been zero till air continuity was reached at the emergence point
saturation which was reached at a lower boundary pressure of −15 cmWC . Resultant pressure conductivity relations for the three models are presented in Figure 5.5
(center). Because of the additional degree of freedom of the EP model, it fitted
measured data best but the slope of the measured curve was much steeper than
the model predicts which might have been caused by the discrete pressure steps.
Hysteresis of air conductivity was reflected by all three models. Deviations became
more obvious in the linear saturation conductivity relation (right). The air conductivity data showed strong hysteresis, which was predicted only slightly by the
models. The best fit was achieved for the EP model with a rescaling of both, the
emergence point water saturation, which replaces the maximum saturation in the
VGM model, and the residual saturation. The dotted lines in the figure show a
g replacementssimulation
PSfrag
replacements
of airreplacements
conductivity with SPSfrag
and Sw,r = 0.7 for the drainage and
w,e = 0.8
Sw,e = 0.5 and Sw,r = 0.4 for the imbibition branch. With the replacement of
the residual water saturation the effect of a constant maximum air conductivity,
ctivity
relative
[-] conductivity
relative
[-] conductivity [-]
S [-]
S [-]
S [-]
Sw [-]
Sw [-]
Sw [-]
[ cmWC ] log ψ [ cmWC ] log ψ [ cmWC ]
1.0
1.0
G
0.9
0.8
0.4
F
0.3
P
0.2
D
0.1
0
0
1
psi
vG
VGM
EP
VGM
VGB
EP
2
0.5
0.4
F
0.3
P
0.2
D
0.1
0
0
1
psi
vG
VGM
EP
VGM
VGB
EP
2
0.6
k
0.5
0.7
0.6
k
theta
0.8
0.7
0.6
G
0.9
0.8
0.7
vG
VGM
EP
VGM
VGB
EP
1.0
G
0.9
0.5
0.4
F
0.3
P
0.2
D
0.1
0
1
0
thetaw
Figure 5.5: Hysteretic pressure saturation relation (left), pressure-conductivity (center) and saturation-conductivity functions (right) for water and air. The
VGM and VGB models are based on the same pressure air saturation
relation. The EP model calculates an effective air saturation which depends on the emergence point saturation. Experimental data measured
in sample S-II with drainage (4) and imbibition (5) branch of the primary drainage curve.
92
5.1 Characterization of Homogeneous Sands
5
0
-5
2
bc
outflow
-10
-15
-20
1
-25
-30
-35
0
0
20
40
60
80
100
120
140
t
100
5
80
4
60
sat
h
6
3
40
2
20
1
0
PSfrag replacements
20
40
60
80
100
120
140
t
700
EP
600
VGM
500
VGB
k
e outflow [cm]
ssure [ cmWC ]
ctivity [cm/h]
time [h]
height [cm]
VGM
VGB
EP
saturation [%]
0
0
400
300
200
100
0
0
20
40
60
80
100
120
140
t
Figure 5.6: Simulation of air conductivity in S-II sand column. Top: Measurement
and simulations of hydraulics. Center: Temporal development of saturation in a vertical profile of the sample. Bottom: Air conductivity calculated for three air conductivity models. Nodes with saturation more
than emergence point saturation are framed black.
93
5 Two Phase Flow Measurement with repacked Sand Columns
measured in the experiment, could be considered in the macroscopic view of these
simulations.
The effect of a vertical saturation gradient due to the influence of gravity was
taken into account when using the discrete evaluation method for air conductivity
introduced in Section 3.6. Development of saturation in a one-dimensional vertical
section of the sand column is shown in Figure 5.6 (center). All model predictions
were based on the hydraulic properties evaluated from the multistep outflow experiment. The VGM and VGB models were not able to describe the data as they used
a continuous smooth saturation function, which was complementary to water saturation. The EP model, which was calculated with the rescaled effective continuous
air saturation had best described the measurement due to its additional degree of
freedom. Emergence point water saturation was set to 0.80 for this simulation which
was an absolute water content of 30 %. Effective air saturation during drainage was
represented well by the model, therefore the rise of air conductivity could be described. Loss of air continuity in the imbibition branch had been much faster than
predicted by all models. Although the EP model was able to describe the end of
air-flow through the column, the detailed course of the decreasing air conductivity
curve had not been described. In Figure 5.6 the nodes with water saturation higher
than emergence point saturation are framed in black. One framed node in the profile
means that there was no possibility for air to pass the column, i.e. air conductivity
had not been measured.
Hysteresis in Hydraulic and Pneumatic Properties Observed in the Additional
Scanning Cycles
The simulation of water dynamics was done for the scanning cycles in the same
way as for the first cycle. The two drainage and imbibition cycles were simulated
with the constrained Kool and Parker model. With this model it was not possible
parameter
αd [cm−1 ]
n
Ksat [cm/h]
θs
θr
τ
first D
0.049±0.000
5.55±0.17
15.7
0.38
0.01±0.01
0.84±0.26
first I
second D
0.075±0.001 0.048±0.001
3.72±0.23
5.15±0.38
15.7
15.7
0.38
0.38
0.01±0.01
0.01±0.01
0.15±0.54
1.07±0.28
second I
0.081±0.001
3.23±0.20
15.7
0.38
0.01±0.01
0.95±0.34
Table 5.4: Van Genuchten parameters for the scanning cycles in homogeneous S-II
sample as estimated with eshpim and their respective 95 % confidence
intervals. Each branch of the curve had been evaluated by separate simulations of the measured curve. The value of θs was fixed for simulations.
∗
Parameter reached limit. D: drainage, I: Imbibition
94
5.1 Characterization of Homogeneous Sands
to fit the measured data. The simulations by sections, where each simulation was
started with the initial pressure and saturation distribution simulated in the former
branch resulted in better agreement between measurement and fit. For the estimations the initial parameters had been chosen with the prior knowledge of the main
branches. The saturated hydraulic conductivity was fixed to the value estimated by
the primary drainage branch and saturated water content was fixed to the measured
porosity.
Based on this preconditions, the scanning cycles were analyzed with a complete
set of parameters for each branch of the curve. The fitted parameters are listed
in Table 5.4 while the resulting hydraulic functions are shown in Figure 5.7. The
additional drainage branches resulted in retention curves similar to the primary
curve. Since the sample was almost resaturated after the first drainage/imbibition
cycle, the parameters should not differ significantly. In the additional imbibition
branches there were only slight differences in the estimated parameters. The different
pressure schemes applied during the scanning cycles had nearly no influence on the
parameter estimation. The resulting parameters estimated for the second scanning
cycle were within the margin of the errors of parameters for the first scanning cycle
although they differed in the number of pressure steps.
The increase of air entrapment after the cycles did not affect the air conductivity
measurement. The emergence point water content which was 16 % in the first cycle,
depended on the applied lower boundary water pressures and the resulting outflow
of water. In the first scanning cycle the measured value was 15 % and in the second
25 % which was due to the large pressure step after 216 hours. The extinction point
water saturation also depended on the lower boundary pressure scheme. In the
primary cycle a water content of 28 % was measured, which was 23 % in the first and
30 % in the second scanning cycle due to the respective responses to the boundary
condition.
The measured air conductivity - saturation relation did not differ significantly in
the three cycles, which made a detailed analysis of the scanning cycles not necessary.
The maximum value was the same for all cycles and corresponded to the value
measured in the oven dry sand.
The analysis of hysteretic air flow behavior during the scanning cycles would
have required lower boundary conditions where the impact on both phases had
been considered. For hydraulic purposes the steps should have varied in height
and length and for comparison of pneumatic measurements the lower boundary
pressure scheme should have been the same. A compromise could have been made
by different pressure schemes which should have been repeated to see the effects on
both phases.
95
5 Two Phase Flow Measurement with repacked Sand Columns
Sfrag replacements
PSfrag replacements
L0
L0
16
0.35
L1
L1
14
0.30
12
L2
L2
θ
vity [cm/h]
ψ[ cmWC ])
10
L3
0.20
0.15
k
theta
0.25
θ
conductivity [cm/h]
log(ψ[ cmWC ])
L4
0.10
L3
8
6
L4
4
L5
0.05
0
0
L5
2
0
1
psi
1
0
psi
Figure 5.7: Hysteresis in the hydraulic functions of the homogeneous sand S-II measured in a primary (blue) and two additional scanning cycles (first: green,
second: red). Data evaluation was done by separate fitting of single
branches (drainage: solid lines, imbibition: dashed). Left: Pressure saturation curve. Right: Pressure conductivity curve.
5.2 Structured Sand Columns
Besides the influence of texture on the hydraulic and pneumatic properties there
is also an impact of large scale heterogeneities made by the objects building up a
porous sample. For examination of the influence of structure within the samples
two different sand packings were realized. The influence of ”preferential air paths”
was tested with the vertically structured column S-H-I which is sketched in Figure
5.8. The horizontally structured column S-H-II was designed to show the effect of
”large scale air entrapment”. A sketch of this column is given in Figure 5.11.
For examination of the primary drainage curves, a measurement was always began
with saturated columns. For this purpose the single structure elements had to be
saturated first and then merged to their resulting structure. This was done by
saturating the elements consisting of the S-I sand then cooling them down quickly
to a temperature less than −20 ‰ with dry ice. The fixed elements were shaped and
placed in the water saturated S-II sand which had successively been built around
the structure elements to ensure full water saturation in the entire column. After
setting up the structure the sample was left for 24 hours for thawing and thermal
equilibration. Sinking effects due to thawing of the sample were not detected.
sample (dimensions)
S-H-I(h=8.0 cm, ®=16.8 cm)
S-H-II(h=6.3 cm, ®=16.8 cm)
porosity [%]
38±5
39±5
Ksat [cm/h] Ingredients (S-I; S-II)
120
97 %; 3 %
20
84 %; 16 %
Table 5.5: Material properties of the heterogeneous sand columns.
96
5.2 Structured Sand Columns
T
10 mm
60 mm
10 mm
168 mm
80 mm
tensiometer
L3
L5
PSfrag replacements
L2
L1
L4
Figure 5.8: Packed sand with predefined structure – S-H-I. The major part (97
Vol. %) consists of sand S-II (light grey) with a ”preferential air path”
and an inclusion consisting of sand S-I (dark grey). The tensiometer is
placed above the S-I sand lens.
5.2.1 Vertically Structured Sand Packing – S-H-I
With the method of vertical air-flow measurement the influence of continuous structure elements in vertical direction could be observed in heterogeneous samples. For
this purpose, the structured column, S-H-I, was built which consisted mainly of the
S-II sand (97 %) with a vertical cylinder (1 cm in diameter) consisting of the S-I
sand, which is placed in the center of the column. An additional S-I lens was placed
at one side of the column for the additional effects of a ”macroscopic dead end”. The
large coarse textured core should simulate the effect of macropores which are often
found in natural soil. The tensiometer was placed one centimeter below the upper
boundary above the coarse lens. Geometric details are shown in Figure 5.8.
Considering the material properties of the homogeneous sands the boundary conditions for this experiment were chosen with the aim of draining the core immediately. The lens should be drained next and then the S-II sand should be drained.
The applied pressure scheme is shown in Figure 5.9.
Experimental Results
The results of the multistep outflow measurement are shown in Figure 5.9. Air entry
pressure had been reached within the first pressure step of −5 cmWC as expected for
the S-I sand. In the second and third drainage step outflow was very slow although
tension at the tensiometer followed the lower boundary pressure immediately. This
effect was caused by the dead end position of the coarse lens, which was drained
through the S-II sand because the core was already drained. At a lower boundary
pressure of −45 cmWC the sample was almost dry. The saturation of the reversal
point was kept at its maximum value until the lower boundary pressure had become
−15 cmWC . At the end of the measurement there was about 10 % of residual air in
97
5 Two Phase Flow Measurement with repacked Sand Columns
1000
0
3
800
-10
600
2
-20
L2
L3
L4
400
rag replacements
time [h]
utflow [cm]
ity [cm/h]
e [ cmWC ]
-30
1
200
-40
0
0
0
10
20
30
L1
40
50
60
Figure 5.9: Multistep outflow measurement results for structured sand S-H-I. Outflow curve (symbols), lower boundary pressure (blue, dashed), tensiometer potential (green), and air conductivity (red) measurements. The
solid lines represent fitted simulation results for outflow and tensiometer
curves.
the sample, which could be related to textural effects and additionally to the coarse
textured structure elements, especially the dead end lens which was enclosed during
the last imbibition step.
With drainage of the vertical cylinder, a continuous air path through the column
had been opened and air conductivity increased to its maximum value within one
hour in the redistribution phase of the −5 cmWC pressure step. The maximum
value was kept during the whole experiment. The extinction of air conductivity
could not be measured, because the air-flow measurement had to be stopped at a
lower boundary pressure of ±0 cmWC to ensure phase separation. At this pressure
the coarse textured S-I sand had not yet been saturated, i.e. only a closed water
table would have stopped vertical air-flow through the sample.
Hydraulic properties
For this kind of pathologic structure evaluation needed a combination of separate fitting of both branches of the measurement and the more flexible mulitmodal hydraulic
functions. Because of the vertical structure the system had been ideally conditioned
for a bimodal model, as two materials were involved, they were vertically arranged,
98
5.2 Structured Sand Columns
and both were directly connected to the water and air reservoirs.
The fitting routine in eshpim only allowed the adjustment of saturated conductivity, residual water content and tortuosity. The other parameters were fixed to
the values estimated for the homogeneous materials. The parameters for the entire primary drainage/imbibition cycle resulting from this evaluation are shown in
Table 5.6. The fitted outflow and tensiometer potential curve is also presented in
Figure 5.9 (solid lines). The simulation results showed good correspondence to the
measurement in the outflow curve because this setting of the column offered the
best conditions for the use of this type of simulation. In the imbibition branch of
the simulated curve correspondence to measured data was still good. Residual air
had been overestimated by the model but this was a result of the specific sample
geometry. The S-I sand cylinder in the middle of the sample offered a central air
conduit. Air from the S-II sand could be removed efficiently via this path over the
whole height of the column until it was saturated in the last pressure step. In this
context a good fitting result always have to consider the fact that the α’s and n’s
were fixed to values estimated from separate experiments.
Pneumatic Properties
For this type of structure, the concepts of air conductivity models discussed in
Section 2.3.4 did not work. Figure 5.10 shows the application of the models with the
same preconditions used for the S-II sand column (Section 5.1.2) as solid and dashed
lines for drainage and imbibition, respectively. Water saturation was represented
by the bimodal hydraulic functions with the 97 % weight for the S-II sand. Air
parameter
drainage imbibition
α1 [cm−1 ]
0.052
0.082
5.88
3.15
n1
−1
α2 [cm ]
0.078
0.142
n2
6.76
6.76
weight 1:2
0.97
0.97
Ksat [cm/h] 118.5±0.6
118.5
0.38
0.38
θs
θr
0.09±0.01 0.05±0.01
τ
0.70±0.15 1.92±0.02
Table 5.6: Van Genuchten parameters of heterogeneous S-H-I sample as estimated
with eshpim and their respective 95 % confidence intervals. Two involved
materials were taken into account by bimodal hydraulic functions. Hysteresis was considered by separate simulation of drainage and imbibition
branch of the measured curve. The value of θs , and the weight of the two
modes of hydraulic functions were fixed for simulations.
99
g replacements
PSfrag replacements
PSfrag replacements
VGM
VGB
EP
1.0
0.9
G
0.8
0.8
0.7
0.7
vG
0.6
VGM
EP
0.5
0.4
F
0.3
P
0.2
D
0.1
0
0
1
psi
VGM
VGB
EP
2
1.0
G
0.9
0.8
G
0.7
vG
0.6
VGM
EP
0.5
0.4
F
0.3
P
0.2
D
0.1
0
0
1
psi
VGM
VGB
EP
2
0.6
k
VGM
EP
1.0
0.9
k
vG
theta
relative
ctivity [-]air conductivity
relative[-]air conductivity [-]
5 Two Phase Flow Measurement with repacked Sand Columns
S [-]
S [-]
S [-]
Sw [-]
Sw [-]
Sw [-]
[ cmWC ] log ψ [ cmWC ] log ψ [ cmWC ]
0.5
0.4
F
0.3
P
0.2
D
0.1
0
1
0
thetaw
Figure 5.10: Comparison of different air conductivity models and data measured in
heterogeneous column S-H-I. Pressure saturation relation for water and
air (left). The hysteretic pressure air-conductivity (center) and saturation air-conductivity (right) functions measured in sample S-II with
drainage (measured: 4, modeled: solid) and imbibition (measured:
5, modeled: dashed) branch of the primary drainage curve. Bimodal
modeling was compared to an unimodal model based on the S-I sand
(dotted).
conductivity had been calculated with the effective saturations and the bimodal
functions for the three models. The models had predicted a slow increase of air
conductivity which was in contrast with measured data.
The direct evaluation of the measured macroscopic data was improved, when the
sample was divided in two distinct pore systems. The S-II sand which had almost
no influence on the air conductivity measurement and the S-I sand which could have
been seen as a macropore system. Since the air-flow kept its maximum value during
the whole imbibition branch this type of evaluation was carried out for the drainage
branch only. The effect of the preferential air path was considered by simulation
of pneumatic conductivity based on the material properties and the portion of the
S-I sand. When air conductivity was calculated only for this part of the sample
the simulations fitted measured data better. The results for the drainage branch
are shown in Figure 5.10 as dotted lines for the three air conductivity models. The
slope of the curve could not be predicted by the models. The steeper increase of
measured air conductivity might be a result of the integral measurement values of
lower boundary pressure and saturation which were taken for data evaluation. For
the ka (ψ) data-points every step was reflected by one distinct water phase pressure
which was correlated to an air conductivity value. The result were discrete points
of air conductivity measurements which caused a steeper increase of the curve. The
ka (S) relation was considering the saturation of the whole column consisting of the
S-I sand, which brought little errors since the S-II sand might have been drained
100
5.2 Structured Sand Columns
parallel to the S-I material. Even though, both simulations showed clearly the
necessity of a separate simulation of the coarse textured core and the remaining
part of the sample.
The distinct influence of the structure of this column with the vertical orientation
made the simple concepts of averaging of pneumatic properties obsolete for this
sample. The concept of emergence point saturation as a trigger for air conductivity
was invalid in this case, as the air used the paths where resistivity was low and
passed the sample. This effect had been seen as preferential air-flow in analogy to
terminology used in solute transport problems where comparable structure elements
are a shortcut for solutes. In the same manner, the coarse textured cylinder acted
as a preferential flow path for air in this sample.
Hysteresis in Hydraulic and Pneumatic Properties
The hysteresis in water dynamics has been represented well by the separate sets
of parameters for drainage and imbibition. For the given structure the bimodal
model could be applied successfully and measured data could be fitted for the whole
drainage/imbibition cycle with the known properties of the homogeneous materials.
The influence of the separate fitting on the saturation conductivity relation are
shown in Figure 5.10 (right) where the hysteretic hydraulic conductivity curve is
also shown.
Pneumatic measurement showed strong hysteresis which could not be represented
by any of the three models. Effective air conductivity of the coarse cylinder depended
strongly on the history of measurement. The sample had only two states which could
be detected with the measurement method used. For full water saturation there was
no air conductivity measurable. The second state started with desaturation of the
coarse structure element as air conductivity reached its maximum value immediately.
The influence of the S-II sand on air conductivity was displaced by the strong impact
of the S-I core.
The measurement of this type of structure required exact knowledge of the initial saturation of the whole sample, since hysteresis had a strong impact. The way
of saturating the sample was crucial for the experimental success, since saturation
by mere capillary rise could have left the ”macropores” open and the experimental
result would have been different. However, these effects can be used for the analysis of macropore systems, their pore continuity, tortuosity, and other geometric
factors.
101
5 Two Phase Flow Measurement with repacked Sand Columns
L2
L1
PSfrag replacements
L4
L3
20 mm
40 mm
63 mm
96 mm
168 mm
L5
tensiometer T1
tensiometer T2
L6
L7
Figure 5.11: Packed sand with predefined structure – S-H-II. The major part (84
Vol. %) consists of Sand S-II (light grey) with a lens (16 Vol. %) consisting of Sand S-I (dark grey).
5.2.2 Horizontally Structured Sand Packing – S-H-II
The influence of structure on air entrapment effects was demonstrated with the
horizontally structured column S-H-II. A coarse textured S-I lens was embedded into
the fine S-II sand. Figure 5.11 shows the setting of the sample and the positions of
two tensiometers. For the measurement of this sample, a small (3.5 mm in diameter)
tensiometer was placed directly in the lens and a second tensiometer at the upper
part of the column in the fine sand.
Boundary conditions were chosen with respect to measurement with the homogeneous S-II sand, which had been the dominating material for this experiment. A
second demand was the availability of air conductivity measurement, accessible at
pressures below ±0 cmWC . The applied air pressure gradient was 4 hPa.
Experimental Results
Measured curves for this experiment are shown in Figure 5.12 (top). When the air
entry was reached, at a lower boundary water pressure of −10 cmWC the outflow
curve had a fast rise and reached a plateau. During the next two steps the S-I lens
was drained resulting in a decrease of conductivity. As a result, the outflow curve
had a slow rise as water from the upper part of the sample had to flow around
the coarse S-I lens. The measured tensiometer potential of tensiometer 2, which
was placed in the coarse lens, had been parallel to tensiometer 1 until the lens had
almost dried out after 20 hours. From that point the hydraulic conductivity became
too low to follow the lower boundary pressure until the water content in the lens had
become 12 % after 41 hours. In the imbibition branch, dynamic effects were observed
when the water had flowed back into the sample. The inflow steps showed a fast
exponential response to lower boundary pressure followed by a linear part which was
102
5.2 Structured Sand Columns
2
0
-20
1
bc
outflow
-10
-30
-40
0
0
10
20
30
40
50
t
100
6
80
5
60
sat
h
4
3
40
2
20
1
PSfrag replacements
0
0
10
20
30
40
50
t
700
600
EP
500
VGM
400
VGB
k
time [h]
ve outflow [cm]
uctivity [cm/h]
essure [ cmWC ]
er saturation [-]
he sample [mm]
VGM
VGB
EP
0
300
200
100
0
10
20
30
40
50
t
Figure 5.12: Multistep outflow and air-flow measurement with sand structure S-HII. Top: Outflow curve (black; measured: symbols, fitted: solid line),
lower boundary pressure (red dashed), tensiometer potential in fine
sand (green; measured: symbols, fitted: solid line) and in coarse lens
(cyan). Center: Temporal evolution of a vertical saturation profile.
The black framed pixels have saturations higher than emergence point
saturation used for the EP model. Bottom: Measured air conductivity
(symbols) and application of VGM,VGB, and EP models.
103
5 Two Phase Flow Measurement with repacked Sand Columns
parameter
drainage imbibition
α1 [cm−1 ]
0.052
0.082
n1
5.88
3.15
α2 [cm−1 ]
0.078
0.105
6.76
6.76
n2
weight 1:2
0.16
0.16
Ksat [cm/h]
15.7
15.7
θs
0.39
0.39
θr
0.07±0.02 0.03±0.01
τ
0.67±0.05 0.57±0.12
Table 5.7: Van Genuchten parameters for heterogeneous S-H-II sample as estimated
with eshpim and their respective 95 % confidence interval. Hysteresis was
considered by separate simulation of each branch of the measured curve
and influence of structure was considered by usage of a bimodal model.
The values of θr and θs were fixed for simulations.
a result of air inclusions which were dissolved in the following redistribution phase.
At the end of the cycle, there had been about 5 % residual air in the sample.
Gas phase continuity through the column was matched at a total water content
of 14 % at a lower boundary pressure of −15 cmWC . Conductivity rose in three
steps and reached its maximum value at a water phase pressure of −25 cmWC . In
the imbibition branch conductivity showed hysteresis referring to water saturation.
Continuity of the air phase was lost at a water saturation of about 30 % at the
interim from the −10 to the −5 cmWC step.
Hydraulic Properties
Figure 5.12 (top) also shows the results of the inverse parameter estimation for
the measured outflow curve which had been done with the bimodal model. The
simulations are plotted in the same color as the respective measurement curve. For
simulations, the tensiometer measurements of tensiometer 1 were selected as it was
placed in the continuous S-II sand. For this simulation, the parameters of the
homogeneous sands were fixed and only residual water saturation and tortuosity
was fitted to measured data. Simulation was done separately for both branches of
the measured curve.
The simulation predicted a fast outflow at the beginning of the drainage branch,
caused by the coarse textured fraction. As expected for the horizontal structure
the simulations did not fit the data at the −15 and −20 cmWC steps. This was a
result of the S-I lens which still had no contact to air reservoir. For the imbibition
branch simulations fitted measured data better as long as air could leave the lens.
With resaturation of the S-II sand in the upper part of the column air had been
104
5.2 Structured Sand Columns
entrapped in the lens and simulation results diverge from the measurement. Parameters estimated by these simulations are listed in Table 5.7 together with the fixed
parameters of the homogeneous materials.
Pneumatic Properties
For quantitative analysis of measured air conductivity data, saturation profiles were
determined in two separate ways. The first approach was based on the previous
knowledge of the type of structure and the material properties of the homogeneous
constituents building up the structured sample. A second analysis was carried out
without any preconditioned evaluation of material properties.
Based on the prior knowledge of the hydraulic properties of the homogeneous
materials and the structure of the sample, air conductivity was simulated with a
simplified approach. The vertical saturation profiles were calculated for the S-II
sand as it had the major influence on air conductivity in this sample due to a
continuity of this material and the horizontal alignment of the S-I lens. The central
part of Figure 5.12 shows the temporal evolution of the saturation for a vertical
profile of the sample. Air conductivity was calculated for the VGM, VGB, and EP
model for the simulated saturations. Starting air-flow had been predicted by all
models, but the increase of air conductivity after 11 hours could not be fitted by
the VGM and VGB models and therefore both models mismatched the measured
data for the rest of the drainage branch. The EP model fitted measured data best
as it had the possibility to simulate the fast rise in air conductivity by adjustment
of the additional emergence point saturation parameter. For the imbibition branch,
both the VGM and the VGB models were not able to fit measured data as they
underestimated air conductivity during the whole cycle. The EP model described
the stepwise reduction of air conductivity qualitatively good but did not fit the
data well. The loss of air continuity and therefore the end of the air conductivity
measurement had also been fitted by the additional extinction point parameter.
The underestimation of the maximum conductivity by all models was a result of
the simplification of the evaluation method. The coarse S-I lense had an influence
especially in the dry range of the experiment which had not been taken into account
by this simulation.
For the second evaluation method no prior knowledge of the system was used.
The experimental multistep outflow data was evaluated with the bimodal model
for both branches of the curve. Fitting results are shown in Figure 5.13 (top).
The new van Genuchten parameters which were result of these fits are listed in
Table 5.8. With these sets of parameters it had been possible to fit the multistep
outflow curve well but the model was not able to predict tensiometer measurements
which was due to the assumption of a homogeneous distribution of the two materials
reflected by the bimodal model. The saturation profiles which resulted from this
simulation are shown in the center of the figure. Based on the new set of hydraulic
properties and the simulated saturations the air conductivity models were applied
105
5 Two Phase Flow Measurement with repacked Sand Columns
2
0
-20
1
bc
outflow
-10
-30
-40
0
0
10
20
30
40
50
t
100
6
80
5
60
sat
h
4
3
40
2
20
1
Sfrag replacements
0
0
10
20
30
40
50
t
900
800
EP
700
VGM
600
VGB
k
time [h]
outflow [cm]
ivity [cm/h]
ure [ cmWC ]
aturation [-]
ample [mm]
VGM
VGB
EP
0
500
400
300
200
100
0
10
20
30
40
50
t
Figure 5.13: Independent evaluation of multistep outflow and air-flow measurement
with sand structure S-H-II. Top: Outflow curve (black; measured: symbols, fitted: solid line), lower boundary pressure (red dashed), tensiometer potential in fine sand (green; measured: symbols, fitted: solid line)
and in coarse lens (cyan). Center: Temporal evolution of a vertical saturation profile. The black framed pixels have saturations higher than
the emergence point saturation used for the EP model. Bottom: Measured air conductivity (symbols) and application of VGM, VGB, and
EP models.
106
5.3 Results of Measurement with Sand
parameter
drainage
α1 [cm−1 ]
0.042
n1
6.12
α2 [cm−1 ]
0.088
12.58
n2
weight 1:2
0.83
Ksat [cm/h]
43.0
θs
0.32
θr
0.01
τ
1.05
Packings
imbibition
0.084
2.66
0.053
15.00
0.91
43.0
0.37
0.01
1.55
Table 5.8: Van Genuchten parameters for heterogeneous S-H-II sample as estimated
by an independent evaluation with the bimodal model. Hysteresis was
considered by separate simulation of both branches.
to the temporal evolution of the saturations. The calculated air conductivities are
shown in the lower part of the figure. A general effect which occurred for all three
model is the overestimation of air conductivity. The coarse part of the assumed
bimodal pore system had been drained fast providing continuous paths for air-flow.
Especially for the VGM and the VGB model these simulations resulted in nonzero air
conductivity in the first 8.5 hours of the experiment where no air-flow was detected.
The same effect was observed at the end of the air-flow measurement where the
models predicted a nonzero flow which was not measured. The fast changes of
conductivity was best described by the EP model, but maximum air conductivity was
also overestimated by this model. Altogether, the deviations between the predicted
air conductivities and measured data were higher than for the simplified approach,
with the known material properties and geometry although the hydraulic data was
represented better by the bimodal hydraulic model.
5.3 Results of Measurement with Sand
Packings
For the investigation of artificially structured samples, precise knowledge of the hydraulic and pneumatic properties of the single materials building the sample was
necessary. Modeling of measured data with the Kool and Parker model combined
with the van Genuchten parameterization for hydraulics and EP model for pneumatics resulted in best description of the single materials. Simulation results were
able to represent measured data. For both, the S-I and the S-II sand the major
deviations were in the wet and the dry range of the measurement whereas the rest
was predicted well by the model. In the wet range an explicit multiphase description
107
5 Two Phase Flow Measurement with repacked Sand Columns
would have been necessary since the assumptions made for the Richards equation
might have been violated. In the dry range the parameterization did not offer
enough flexibility for desrciption of the system. The observed deviations resulted in
similar differences between measurements and simulations of the structured samples
where the simulations were based on the material properties of the homogeneous
materials.
With the fixed hydraulic properties of the homogeneous materials the simulations
of the structured samples, required an adequate model. For the S-H-I sample the
bimodal van Genuchten model was able to simulate measured data well, because the
sample geometry met all requirements for this type of model. Pneumatic behavior
could not be modeled for this structure as a whole. The models used for simulations
did not reflect structure. For this case air conductivity had to be separated for
all continuous structure elements. Air conductivity had to be modeled for every
compartment and summed up for the whole sample to get the effective conductivity.
With the knowledge of the properties of the compartments, the method can be
applied for detection of vertically continuous macropore systems.
Hydraulic properties of the heterogeneous S-H-II sample could not be described by
the bimodal model for the outflow branch as the coarse lens had no direct connection
to the air reservoir during drainage. With the reduction to the fine textured material
the pneumatic behavior could be described well. A second approach where simulations were carried out without reflecting the prior knowledge of material properties
showed the connection of both the hydraulic and the pneumatic material properties.
For this type of structure explicit multidimensional simulation of the water and
air dynamics of the multistep outflow and air-flow experiments would be necessary.
Current simulation codes are able to use both, the van Genuchten and the Brooks
and Corey parameterization (Ippisch 2001), and the more flexible type of cubic
spline functions for description of the hydraulic parameters. A combination of these
forward simulation programs with an adequate inversion algorithm should make
parameter estimations possible, where only the structure of the sample and the
multistep outflow data will be necessary.
108
6 Conclusions
In this study multiphase investigations were carried out on different types of porous
media. For measurement of air conductivity, which is an important additional material property, an experimental device had been developed which allowed measurements of both, the hydraulic and the pneumatic state variables. The device for transient measurement of hydraulic properties within multistep outflow experiments was
combined with an air permeameter which allowed simultaneous determination of air
conductivity. The special lower boundary plate which was necessary for separation
of phases, offered an interface to the further measurement instruments. The experimental setting was applied to two different types of porous media. Artificial sintered
glass columns, a highly idealized porous media and several packings of sieved sands,
representing more natural media, were examined, using the multiphase investigation
approach. Although it has not been tested yet, the setup should be also suitable
for the measurement of undisturbed soils. The multistep outflow part of the setup
has already been tested with soil cores and with an appropriate sealing at the side
boundary, it is also possible to do the additional airflow measurement.
In the case of the sintered glass material the connection between homogeneous materials and heterogeneous samples had been examined. Hydraulic properties of homogeneously textured columns were estimated from multistep outflow experiments.
Measured data had been modeled inversely using the van Genuchten-Mualem parameterization with the constrained Kool and Parker approach for hysteresis. Fitting
results were only satisfactory for the coarse textured material. The fine textured
sintered glass required more flexibility in hysteretic hydraulic functions to describe
hysteresis phenomena. For this purpose the constraint of a single conductivity function for drainage and imbibition of porous media has been released and a better
description of hysteretic hydraulic functions has been achieved by the Kool and
Parker model.
Measurements with a heterogeneous column, which consisted of the investigated
homogeneous materials, showed important effects of structure on hydraulic and
pneumatic properties. The horizontally structured sample could not be simulated
with a multimodal model, when the hydraulic parameters were fixed to the values
evaluated for the homogeneous samples. This was expected, as not all compartments
had direct connection to air and water reservoirs throughout the whole experiment.
For simulations of horizontally structured samples it would have been necessary to
109
6 Conclusions
use three-dimensional models based on explicit sample geometry. Furthermore there
was a lack of accurate description of the hydraulic properties of homogeneous materials, as even small deviations especially in the wet and dry range lead to wrong
simulations of structured samples. The van Genuchten model, with its low number
of parameters and its constraint, which reduces flexibility of the function, failed
especially in these ranges of saturation.
Air-flow measurements showed a sharp breakthrough of air and an increase of air
conductivity due to the decreasing water saturation of the heterogeneous sample.
For simulations of air conductivity it was satisfactory to simulate discrete water
saturation profiles for the finer textured homogeneous material. The simulations
showed that the major influence was by the lower layer of the sample which was the
homogeneous lower third of the column. Continuity of air depended only on saturation of this finer material, which was continuous throughout the whole sample.
Applied air permeability models described data qualitatively well. Quantitative description was best for the EP model. Simulation of air conductivity with the concept
of an emergence point saturation, which was necessary to establish a continuous air
phase, was the appropriate model for measured data. The VGB and VGM models
which were based on a continuous air phase saturation overestimated air-flow in the
wet range. Differences in the predictions of these models could be explained by the
different concepts of the pore network system.
The concept of the transfer of hydraulic and pneumatic properties of homogeneous
materials to heterogeneous samples was successfully demonstrated with the repacked
sands. Estimation of hysteretic hydraulic properties for the coarse homogeneous
sand was done with the constrained Kool and Parker model. The fine textured
sand required the more flexible model. Resulting hysteretic hydraulic functions did
not contradict physical principles within the measurement range. Measurement of
additional scanning loops confirmed the evaluation method. Modeling of air-flow
measurements based on forward simulations with estimated hydraulic properties
was best with the EP model. Here, the emergence and extinction point saturations
were almost constant for the scanning cycles.
Investigations with structured columns clearly showed the effect of the orientation
of structure elements. For simulations of hydraulic and pneumatic properties of the
structured samples, all information available from homogeneous materials was used.
For the vertically structured sample a multimodal model could be used for description of hydraulics, but all air-flow models failed if both materials were considered
with their respective shares as the coarse structure element acted as a preferential
flow path for air. The separation of this macropore system led to better results in air
conductivity predictions, if the simulations were based only on the coarse textured
material.
The influence of the different horizontal layers within a sample could be shown
with a second structured column. One-dimensional modeling of air conductivity
could be based on the continuous fine sand only where the influence of the coarse
110
lense became visible only in the dry range. A comparison of two different simulations of this experiment showed the connection between hydraulic and pneumatic
properties. A bimodal best fit of hydraulic data without any prior knowledge of
the sample, could not fit the measured air-flow data, since the model did not reflect
the influence of structure. The pressure-saturation hysteresis for this sample was
causally connected to processes at two distinct scales. The residual air at the end
of the multistep outflow experiment could be assigned to both the texture of the
materials, where capillary effects were dominant, and additionally to the structure,
where the coarse lense was embedded in the fine material.
The experiments showed that sample structure had an enormous impact on combined measurements. Effects which could not have been seen with a multistep
outflow experiment were observed by the additional air-flow measurement. The vertical structure resulted in little outflow of water but had a fast breakthrough of air.
Continuous preferential flow paths in vertical direction were found in fast experiment over a wide pressure range. Classically they have been detected by transport
experiments at one specific saturation. For scanning a specific pressure range several
tracers had to be used in very time consuming experiments. The horizontally structured sample showed the necessity for multidimensional simulations where enclosed
structure elements could be represented. The bimodal model which was used was
not able to predict the experimental results.
Determination of unsaturated air conductivity from hydraulic properties required
a rescaling of the effective saturation as proposed by the EP model. The fast rise in
air conductivity with decreasing water saturation at a specific emergence point, to
a maximum value which is almost fixed over a wide range of saturation could be described best by a rescaling of both, the maximum and the residual water saturation
parameters. Although both fluids share the same pore space the effective saturations of both could not be seen complementary. The experimental evidence for the
different effective pore systems should be incorporated into future air conductivity
modeling.
111
6 Conclusions
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122
Appendix
123
.
124
A Setup
A.1 Pressure Sensors
The sensors used are from the Honeywell1 PC26 series with the technical features:
model
mode
pressure range
max. overpressure
compensated
response time
linearity
typical sensitivity
null offset
repeatability and
hysteresis error
operating temperature
range
26PCBFA6G
absolute
±5.0 psi
20.0 psi
yes
1 ms max.
0.40% span. typ.,
0.50% span. max.
10 mV/psi
0 mV typ.
0.20% span. typ.
26PCAF6D
differential
±1.0 psi
20.0 psi
yes
1 ms max.
0.25% span. typ.,
0.50% span. max.
16.7 mV/psi
0 mV typ.
0.20% span. typ.
-40 ‰ to 80 ‰
-40 ‰ to 80 ‰
A.2 Experimental problems
The rigid structure was one of the main features provided by the sintered glass
columns. This advantage of the material was necessary during the test stage of the
experimental setup as the experiments had been redone several times for development of the setting and exclusion measurement artifacts2 . Some of these artifacts,
as well as, if possible, a reason for the removal of their effects should be documented
1
2
www.honeywell.com
A fixed medium offers the best possibility for comparison of experiments carried out at the
same setup. Differences in measured data between two measurements with the same boundary
conditions has to be artifacts from anywhere in the whole measurement procedure.
125
A Setup
in this extra section, as they may help other scientists avoid some troubles or even
give focus to details, which are not obviously troublesome, but which may cause a
lot of confusion.
ˆ Never use any random type of glue for fixing porous glass plates. Some glues
shrink a little while they harden and the developing shear forces can kill the
glass plates.
ˆ A major problem for gas flow experiments is the sealing of the samples. For
this study several casings were tested for the glass samples. Latex casings were
too weak and widened with increasing water pressure, resin was drawn in the
pores of the sample and reduced the available porosity, additionally there is the
problem of some vapor of a solvent which might influence surface properties.
Best results were achieved by shrinking PVC tubes. They produced no surface
active vapors and when fixed onto the samples they were rigid within the
required pressure range.
ˆ Never use silicone in the whole setup, as the vapor is an highly active surfactant, which repelles water efficiently and makes the porous material hydrophobic. This is true for all kinds of sealing such as lower or side boundaries,
tensiometers, and TDR.
ˆ As a soil column may have a weight of a few kilograms, the separator plate
must be able to withstand this. The first approach, using a plastic grid glued in
the funnel below the plate failed, as it bent till the plate cracked. A monolithic
design where the plate holder and the supporting grid were milled from a single
piece of plastic was much stiffer.
ˆ The optimal way to calibrate the pressure sensors for tensiometer and lower
boundary pressures was a two step procedure. In the first step the slope of
the sensor (cm water-column vs. digit output) had been measured with a
hanging water-column and in the second step the offset was fixed when the
sensor was installed and the applied pressure is well known, e.g., in a fully
saturated sample, the tensiometer pressure at a specific depth is well known.
ˆ One of the major problems in porous plate experiments is the diffusion of
air below the plate. As it is not at all possible to suppress this effect, it is
necessary to have an insight on the funnel below the plate in order to monitor
this effect and to convince its influence on the experiment. I quit using deaired
water, since this was only effective in the initial state of the experiment. After
some scanning cycles the water was exposed to air and mixed again.
ˆ The hydraulic contact between the porous plate and the sample has to be at
a maximum level. To reach this, a contact material was used for the rigid
126
A.2 Experimental problems
media. The material must be thin enough not to influence the experiment,
its hydraulic permeability larger than the investigated material, and it has to
be permeable to air. For these purposes a gauze had been the most useful
because there were fine textured filaments for the water and large wholes for
the air.
ˆ Since I forgot to poison the samples in the beginning some type of bacteria
or other, grew there. Cleaning the samples with acetone, sulphuric acid, and
caustic soda aided in removing these organisms in the columns and caused an
increase of saturated conductivity to the values measured in the first tests.
ˆ Different bubbling outlets for the Mariotte Bubble Tower were tested. On the
one hand the smaller they were the better the resolution of the measurement
but on the other a small jet was of larger resistance and the applied pressure
had to be adjusted higher.
This list may be an incentive for scientists to do the same as there is not enough
time for everyone to make the same mistakes him- or herself.
127
B Sintered glass
The sintered glass material, used for the investigations in Chapter 4 should be briefly
characterized. The borosilicate glass 3.3 is specified as a standard glass - it contains
mainly sand, calcium carbonate and sodium carbonate. Actually this material is
taken for several filter purposes. The technical data provided by the manufacturer
is given here.
B.1 Chemical composition
The following is a typical analysis of borosilicate glass 3.3:
Element
Silica (Si02 )
Boric oxide (B2 03 )
Sodium oxide (Na2 0)
Alumina (Al2 03 )
Iron oxide (Fe2 03 )
Calcium oxide (Ca0)
Magnesium oxide (Mg0)
Chlorine (Cl)
% by weight
80.60
12.60
4.20
2.20
0.40
0.10
0.05
0.10
There may exist heavy metals in a concentration of less than 5 ppm.
B.2 Physical Properties
Coefficient of Expansion
Specific Heat
Thermal Conductivity
Density
Poisson’s Ratio
Young’s Modulus
Rigidity Modulus E
DPH (Vickers) Hardness
Relative Hardness
Refractive Index
Dielectric Constant
128
33·10−7 /‰ between 20 ‰ - 300 ‰
750 J/kg‰ at 20 ‰
1.13 W/K·m at 20 ‰
2.23 x 103 kg/m³
0.22 between 25 ‰ - 400 ‰
6.500 kg/mm² at 25 ‰
62,5 kN/mm²
580 Kg/mm² with 50 gram load
1.52 (comparative Soda-Lime = 1.0)
1.474 Sodium D - Line
4.6 at 1 MHz and 20 ‰
B.3 Influence of Sample Structure on Transport Processes
B.3 Influence of Sample Structure on Transport
Processes
The structure of the heterogeneous sintered glass sample was build with respect to
the influence on water and solute transport at different states of saturation. Numerical simulations with the two-dimensional flow simulation code swms (Šimu̇nek
et al. 1994) were carried out to visualize the effect. The hydraulic properties of the
homogeneous materials were set as estimated in the experiments (Chapter 4). To
the initially dry samples a permanent flow at the upper boundary was applied and
the development of the flow field in the column was calculated. When equilibrium
was reached, the saturations, water pressures and the fluxes in the column were
stored. Simulation of solute transport was done by a particle tracker where the formerly calculated hydraulic states were used. Based on the flow field 10.000 particles
were introduced at the nodes of the upper surface. The propagation of the particles
in time was based on the stochastic velocity field generated by swms and an additional random walk took molecular diffusion and subscale hydrodynamic dispersion
in consideration. The ways the particles moved through the samples were tracked.
Concentrations were calculated by dividing the number of particles in a small region
by the volume of that region (Roth 1996, p. 106).
The results of simulations are shown in Figure B.1 for a low flow rate at the
upper boundary. The applied flow was 10−12 m3 /s and simulation was calculated
for 1010 s. The upper part of the figure shows the saturation for the steady state
(left), the pressure inside the column (center), and the flow field (right). The results
of particle tracking are shown in the lower part where the concentration is shown for
three travel times to visualize the development of the particle plume in the flow field.
The plume was divided into two parts. One part had already passed whereas the
other part was stored above the coarse lens. These particles were transported very
slowly since they had to move around the lens where the flow rates were low.
For the high flow rate (10−2 m3 /s) the steady state was reached after 108 s. Figure
B.2 shows the saturation, pressure, and flow field for this case. In the saturation plot
the lenses cannot be identified anymore since the entire column was fully saturated.
The lower part of the figure shows the shape of the tracer plume which is totally
different from the first case. On its way down the plume was only shaped by the
different conductivities of the lenses but kept together.
129
B Sintered glass
theta
head
flux
0.10
depth
0.08
0.06
0.04
Sfrag replacements
ater content
depth z [m]
x [m]
re head [m]
|jw | [m/s]
e PDF [m2]
0.02
0
0
0.020
0.040
0
0.020
x
1
0.2
6
0.3
1
0
0.020
0.3
6
0.4
0
0.4
-0.
1
5
9
0.040
x
16 0.13 0.10 0.06 0.03
-0.
8
5
4
2
1
11 .261 .059 .135 .301 .665
0
7e
4e
3e
2e
9e
-14 -13 -11 -10 -9 e-8
2.6
depth
0.2
0.040
x
Sfrag replacements
ater content
depth z [m]
x [m]
re head [m]
|jw | [m/s]
e PDF [m2]
92.4 h
184.7 h
277.1 h
0
0
0
0
x
x
31
2
78
7
x
50
19
00
84
12
59
9
31
74
8
dist
Figure B.1: 2-d simulation of water dynamics in the heterogeneous sintered glass
column at a low flow-rate. Top: saturation (left), pressure (center), and
flow field (right). Bottom: Temporal evolution of the particle plume.
130
B.3 Influence of Sample Structure on Transport Processes
theta
head
flux
0.10
depth
0.08
PSfrag replacements
0.06
0.04
ater content [-]]
depth z [m]
x [m]
essure head [m]
|jw | [m/s]
ance PDF [m2]
0.02
0
0
0.020
0
0.040
0.020
1
0.2
6
0.3
1
0.020
0.3
x
6
0.4
0
0.4
1.5
5
3
0.040
x
7.6 13
1
3
2
.75 9.86 5.98 2.09
4
0
0
0
0
0
02 .003 .005 .008 .013 .021
7
6
8
4
4
1
0.0
depth
0.2
0
0.040
x
PSfrag replacements
ater content [-]]
depth z [m]
x [m]
essure head [m]
|jw | [m/s]
ance PDF [m2]
1.2 s
0
2.4 s
0
3.6 s
0
0
x
x
78
19
7
x
12
49
50
6
31
50
79
37
dist
Figure B.2: 2-d simulation of water dynamics in the heterogeneous sintered glass
column at a high flow-rate. Top: saturation (left), pressure (center), and
flow field (right). Bottom: Temporal evolution of the particle plume.
131
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