Piercement structures in granular media

Piercement structures in granular media
Piercement structures in
granular media
by
Anders Nermoen
Physics of Geological Processes
Department of Physics
University of Oslo
Norway
Thesis submitted for the degree
Master of Science
May 2006
“One of the symptoms of an approaching nervous breakdown is
the belief that one’s work is terribly important.”
Bertrand Russel
Acknowledgment
First of all I want to thank my supervisor, Anders Malthe-Sørenssen for all
the inspiring and nice conversations and for not making me feel like a burden.
His deep knowledge and scientific integrity has been very of great help when
I have encountered problems on the way. Thank you!
In the laboratory I wish to thank Olav Gundersen for all the insightful
help and technical support when performing the experiment. I also wish to
thank Sean Hutton for developing much of the experimental setup during
his Post. Doc. period at PGP in 2003. When interpreting the data I will
especially thank Simon deVilliers, Grunde Waag, Berit Mattson and Yuri
Podladchikov for all the fruitful discussions. Thank you!
I would also like to thank the people that have helped me understand the
large picture and the geological relevancy. Especially I wish to thank Henrik
Svensen and Yuri Podladchikov. Thank you!
The last five years can be described by three words; busy, instructive
and fun! I can not acknowledge my fellow students enough for pulling and
pushing me to this point. Without the friendship, the long discussions and
everything you have taught me, the master thesis in physics would never be
finished. I owe you everything.
I wish to thank everyone in “Fysikkforeningen” and “Fysisk Fagutvalg”
and the corporation we had on “Lille Fysiske Lesesal” to reveal the mysteries
in everything from Classical Mechanics to Electromagnetism. To mention
your names would be too risky in case of forgetting anyone, you know who
you are. Thank you!
Secondly I want to thank “The Nice Master Students” on PGP for bringing
joy not only to society in general but also to me personally. It has been a
pleasure to share office with all of you. I wish to thank Torbjørn, Solveig,
Ingrid, Grunde, Helena, Kirsten, Brad and Hilde. Thank you for all the
laughter and fuzz made on the way to this day. Thank you Jostein for all
the discussions on everything from Scotch whisky, via massive neutrinos in
cosmology to which trout flies to chose. I know nobody knowing so many
digits in π and lyrics as you! Thank you!
iii
The environment on PGP is a lot more than the master’s room. I wish
to thank all the employed at PGP for their friendliness and their support.
Without you PGP would not exist. Thank you!
Nobody deserves a paragraph in my acknowledgments as much as my
family. First of all I will thank my parents Allic Lunde Nermoen and Bjørn
Nermoen for their infinite source of loving support through my life. I am not
able to express how much the two of you has meant to me up to this day.
Then I would like to thank my sister Frøydis, for all the nice evenings we
have shared and all the excellent food you make me here in Oslo. To my two
younger brothers, Marius and Jonas, thank you for all the fun we have had.
I hope that we can see much more of each other in the future. To all my my
brothers and sister I will say that I am extremely proud of all of you! My
deepest wish is to keep the closest contact with all of you; Mamma, Pappa,
Frøydis, Marius and Jonas for many years to come. Thank you!
Now in the last paragraph I wish to thank my friends outside of the
studies for forcing me to think of something else than physics. You deserve
my apologies for unjust treatment the last years, to Morten and Morten and
the other members of “The Daggers”. Thank you!
The last one and half year, during the master work, is a period of life I
will look back onto with joy. Thank you to all the happy and nice people
around me for making my life meaningful. Without you nothing could be
done...
Contents
Acknowledgment
iii
I
Introduction
1
1 Physics motivation
3
2 Geological background
2.1 Hydrothermal vent complexes . .
2.2 Kimberlites and kimberlite pipes
2.3 Conditions for venting . . . . . .
2.4 My experiments in this setting . .
II
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Theoretical background
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3 Granular media
3.1 Granular solids . . . . . . . . . . . . . . . . . .
3.1.1 Packing of static granular media . . . . .
3.1.2 The angle of repose . . . . . . . . . . . .
3.1.3 Inter particular forces in granular media
3.2 Force networks . . . . . . . . . . . . . . . . . .
3.2.1 The Janssen law of wall effects . . . . . .
3.3 Mohr circles . . . . . . . . . . . . . . . . . . . .
3.3.1 Failure envelopes/constituent equations .
3.3.2 Coloumb fracture criterion . . . . . . . .
3.3.3 Tensile fracture criterion . . . . . . . . .
3.3.4 Von-Mises failure . . . . . . . . . . . . .
3.4 Granular liquids . . . . . . . . . . . . . . . . . .
3.4.1 Segregation phenomena . . . . . . . . . .
v
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21
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34
CONTENTS
4 Liquid flow in porous media
4.1 Derivation of the NS-equations . . . . . . . . .
4.2 Viscous force . . . . . . . . . . . . . . . . . .
4.3 Reynolds number . . . . . . . . . . . . . . . .
4.4 Euler’s equation . . . . . . . . . . . . . . . . .
4.5 Stokes flow and sedimentation . . . . . . . . .
4.6 Bubble in a viscous fluid . . . . . . . . . . . .
4.7 Darcy’s law . . . . . . . . . . . . . . . . . . .
4.8 Darcy’s law on differential form . . . . . . . .
4.9 Models of permeability . . . . . . . . . . . . .
4.9.1 The capillary model . . . . . . . . . .
4.9.2 Carman-Kozeny model of permeability
4.10 Fluidizing granular media . . . . . . . . . . .
4.10.1 Classical fluidization criteria . . . . . .
III
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Experiment
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5 Venting in the laboratory
5.1 Experimental setup . . . . . . . . . . . . . . .
5.1.1 The material . . . . . . . . . . . . . .
5.1.2 Air supply . . . . . . . . . . . . . . . .
5.2 Performing the experiment . . . . . . . . . . .
5.3 Results . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Linear regime . . . . . . . . . . . . . .
5.3.2 Breakdown of linearity . . . . . . . . .
5.3.3 Fluidization . . . . . . . . . . . . . . .
5.4 Geometrical measurements . . . . . . . . . . .
5.5 Dimensional analysis . . . . . . . . . . . . . .
5.5.1 Dimensional analysis . . . . . . . . . .
5.6 Venting in natural systems . . . . . . . . . . .
5.7 Flow of compressible fluids in granular media
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6 Additional experiments
6.1 Transition from fluidization to fracturing . . . . . . . .
6.1.1 Experiments on dry glass beads . . . . . . . . .
6.1.2 Experiments on a bed of clay . . . . . . . . . .
6.1.3 Experiments on wet bed of glass beads . . . . .
6.2 Heterogeneous beds . . . . . . . . . . . . . . . . . . . .
6.2.1 Experiment with a deep low permeable layer . .
6.2.2 Experiments with a shallow low permeable layer
vi
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59
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90
CONTENTS
6.3
6.2.3 Experiment on two clay layers . . . . .
6.2.4 Experiments of one layer of glass beads
6.2.5 Experiment on large glass beads . . . .
Intermediate cohesion . . . . . . . . . . . . . .
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7 Discussion
7.1 Onset of bubbling . . . . . . . . . . . . . . . . . .
7.1.1 Griffith mode 1 fracture . . . . . . . . . .
7.1.2 Transition from laminar to turbulent flow .
7.2 Onset of fluidization . . . . . . . . . . . . . . . .
7.3 Calculated effective permeability . . . . . . . . . .
7.4 Onset of fluidization, 2. attempt . . . . . . . . . .
7.5 Natural systems . . . . . . . . . . . . . . . . . . .
7.5.1 2D versus 3D modelling . . . . . . . . . .
7.5.2 Piercement structures in nature . . . . . .
7.6 Additional physical effects . . . . . . . . . . . . .
IV
Concluding Remarks
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91
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8 Brief summary and conclusions
119
9 Future work
121
vii
CONTENTS
viii
Part I
Introduction
1
Chapter 1
Physics motivation
The physics of granular media is an interesting field with a lot of research
activity. Granular materials have been studied for over two centuries, though
several of iths rather peculiar properties are still poorly constrained (e.g. it’s
phase diagram). This in contrast to its simplicity; they are large conglomerates of macroscopic particles, and its familiarity to our daily lifes.
Granular materials are known to exist in all three phases of matter. A
pile of sand at rest can be thought of as being in the solid phase. Flowing
particles to some extent behaves as a flowing media, thus it is interpreted to
be in a liquid phase. It is also found in the gas phase1 .
The main problems of understanding and describing the dynamics of fluidized granular media arises due to averaging problems when deriving the
flow equations. The continuum equation is not defined for volumes smaller
than the macroscopic size of the particles. In the other limit, the largest
systems (e.g. corn silos) are far from large enough to be called infinitely
large.
Of great interest these days are the study of the poorly understood phase
transitions in granular materials [1], [2]. Analytical solutions for the phase
diagram of granular materials are of major importance in several fields not
only in physics but also for e.g. city planning2 and geology.
There are at least three ways of fluidizing static granular media. By
tilting a pile of sand over the static angle of repose surface flow of grains
thus fluidization will occur. Secondly by shaking a box of granular media
at frequencies above a specific frequency or by forcing viscous fluids through
the bed the previously static granular media will starts flowing and behave
1
The gas phase of granular materials is not discussed in this thesis though it inhibits
several very interesting properties such as clustering and inelastic collapses.
2
E.g. Landslides and avalanches are common catasthropic events killing people every
year all over the world.
3
CHAPTER 1. PHYSICS MOTIVATION
in a liquid like manner.
In this thesis an experimental study of the transition between static and
viscous flow induced fluidized granular media is presented. The air is injected
through a single inlet at the bottom of a Hele-Shaw cell. A phase diagram
marking the onset of fluidization is presented by measuring the necessary air
flow velocity through the inlet and varying the fill height of the bed.
The thesis is organized in the following manner. In chapter 2 examples
of fluidization and flow localization in geological systems is presented. It
was historically the study of hydrothermal vent complexes that initiated the
study of fluidization and vent formation in laboratory on PGP. A common
feature of the geological examples is that fluidization is induced by high fluid
pressure at depth that initiated an upward viscous flow with velocity v. The
presentation of the geological examples is intended to be readable to readers
with sparse geological training. A brief review of analogue experiments to
model the formation of kimberlite structures and under which conditions
venting occur will be given.
In chapter 3 we give a general introduction to the physics of static and
fluidized granular materials. Several interesting properties of granular materials is presented. The reader familiar to granular materials might skip
this chapter and move on to chapter 4 where fluid flow through porous media is described. In this chapter several concepts such as viscous fluid grain
interactions, Reynolds number, Darcy’s law, permeability and fluidization of
granular media is introduced. The last part of this chapter contains hints on
fluidization that will later be used to analytically explain how the fluidization
velocity depends on fill height.
In chapter 5 the experiment and its results (the phase diagram) is presented. The characterization of the setup, material used, air supply and a description of how the experiment is performed is also given here. We report on
observing three distinct “regimes”; the “linear regime” where normal Darcy
flow applies and we have a linear relation between the flow velocity and the
pressure drop across the bed. Secondly, in the “bubbling regime” a static
stable bubble forms above the inlet causing a break in linearity between the
flow velocity and pressure measurements. And thirdly “fluidization”, where
the bubble rapidly grows to the surface and a vertical conduit forms in the
center above the inlet. The grains are rapidly spouted to the surface through
the vent and a downward flow of grains along the sides defines the fluidized
zone. The onset of these three phases is determined by the inlet air flow velocity. By varying the fill height a phase diagram of the documented features
is presented.
In chapter 6 a brief presentation of several additional experiments is given.
No controlled variation of any physical quantity is done thus no fundamental
4
new physics is gained from these experiments. The experiments are presented since they represent nice examples of; (1) how the competition between
fluidizaton and fracturing depends on grain size and cohesive forces, (2) how
induced heterogeneities effects the fluidized zone, (3) how deformation might
occur in natural settings when low permeable layers are emplaced at shallow
depths, and (4) size segregation of fluidized granular materials.
Chapter 7 contains a discussion and interpretation of the phase diagram.
I addition a discussion on venting in natural systems and additional physical
effects are given.
Chatper 8 contains a brief summary and concluding remarks, while I in
chapter 9 discuss my ideas about the further work related flow localization
in granular media.
5
CHAPTER 1. PHYSICS MOTIVATION
6
Chapter 2
Geological background
This chapter presents two geological examples where fluidization is the key
physical process leading to the formation of the feature seen today.
Piercement structures such as hydrothermal vent complexes (htvc) and
kimberlite crates are in the geology often related to the process of fluidization
of granular media (see Woolsey et. al. 1975 [3], Clement et. al. 1989 [4] and
Jamtveit et. al. 2004 [5]). The formation of these features is related to how
pore fluids are expelled from rocks.
At low pressure gradients the pore fluids within the rock would seep
through permeable rocks by following Darcy’s law. In this model the flow
velocity v is proportional to the pressure gradient ∇p times a prefactor given
by the permeability of the porous bed k, the viscosity of the fluid µ, and the
sample length h, summarized in
v=
k
∇p.
µh
(2.1)
The formations of the piercement structures above are natural examples of
where the Darcy’s law is insufficient to accommodate the imposed flow velocity. Due to gas mass conservation will the pore space break and the flow
will focus through high permeable zones recognized as e.g. the htvc and
kimberlites.
The onset of this focusing process is said to occur when the process of
pressure build up1 happens more rapidly than the process of pressure decrease
(Darcy’s law) [5]. A presentation of this idea is given in section 2.3.
Now in Darcy’s law there is no upper bound for how fast the pressure or
fluids can released through the rock, since there is no physical upper bound
1
There are several ways to build up the fluid pressure at depth. I will give some insight
into two / three examples in the following section.
7
CHAPTER 2. GEOLOGICAL BACKGROUND
of the flow velocity within this model. So to address the case of giving upper
bounds to Darcy’s law we include the process of fluidization. Fluidization
occurs when (1) the gas flow velocity through a bed of particles provides
viscous sufficient drag to lift the overlaying sediments or (2) the pressure at
depth equals the lithostatic pressure of the overlaying sediments. Thus from
(1) at high fluid velocities it is energetically easier to fluidize the bed than for
the bed to get rid of its pressure through Darcy-flow. It is of major interest
to quantify the onset of viscous flow induced fluidization to determine under
which conditions flow localization and venting occurs in nature.
In a recent study of Walters et. al. 2006 [6] an experimental study
of fluid flow induced fluidization was performed. They found that through
fluidization of sand beds, they were able to produce the well defined transition
between the static and fluidized zones that look remarkably similar to those of
kimberlite structures. They suggest that the diverging geometry in kimberlite
pipes occur due to fluidization of mixtures of different sized particles. A
similar experimental study was performed in 1975 by Woolsey et. al. [3] to
evaluate the mechanisms of formation of diatreme structures. They varied
the geometry of the containers and used different sized particles ranging
from clay to 0.5 cm sized gravel. They found that through viscous flow
induced fluidization they could reproduce all the main features seen in Maartype craters2 . Features such as e.g. particle size segregation and cocentric
subsidence around the conduit is recognized within their experiment that
duplicates what is observed in the kimberlite pipes.
The pipe like structures of the hydrothermal vent complexes is also interpreted to be formed through fluidization. A model for the formation of these
features in volcanic basins is identified by e.g. Planke et. al. 2003 [7]. They
identified the following steps of the formation of these features. (1) Intrusion
of magma into sedimentary basins leads to heating and local boiling of pore
fluids within a zone around the intrusion (termed aureole). (2) The increased
fluid pressure causes hydro fracturing due to the formation of Mode 1 cracks.
A fundamental observation of the hydrothermal vent complexes is that they
tend to originate from the tip of the sill intrusion. (3) The fluid decompression leads to explosive hydrothermal eruptions onto the paleosurface, forming
a hydrothermal vent complex. The explosive rise of fluids towards the surface
causes brecciation and fluidization of the sediments and commonly the formation of a crater on the paleosurface. (4) The fracture system created during
the explosive fluidized phase is later re-used for circulation of hydrothermal
fluids during the cooling process of the magma. This stage is associated with
2
A Maar crater is the top feature of Kimberlite structures. They are often seen as
circular lakes at the sorface today.
8
2.1. HYDROTHERMAL VENT COMPLEXES
sediment volcanism through up to several hundred meters wide pipes cutting
the brecciated sediments and rooted as deep as 9 km [8]. (5) At later stages
can the hydrothermal vent complex be re-used as fluid migration pathways
forming seeps and seep carbonates at the surface.
Common for the htvc and kimberlite structures is that they form pipelike
structures. This can be interpreted as being a consequence of the fluidization
of brecciated clasts within a zone. It is the fluidized zone that we see today
as the htvc and kimberlite.
The piercement structures have an important long term impact on the
fluid flow history in sedimentary basins [9], [8]. The high permeable zones
are often re-used for fluid migration during dormant periods and after the
structures become extinct. This is a phenomena that can be deduced from
the hierarchical structure, where smaller pipes are located within the main
pipe structure Svensen (in press) [10].
In the proceeding two sections a presentation of some background studies
of the hydrothermal vent complexes and Kimberlites will be given. Then I
will present a discussion by Jamtveit et. al. in 2004 of under which conditions
venting and break down of the pore space might occur. In the end of this
chapter I will relate and motivate the experimental studies to the geological
observations.
2.1
Hydrothermal vent complexes
Large igneous provinces are characterized by the presence of an extensive
network of sills3 and dykes emplaced in sedimentary strata [10]. Examples
of large igneous provinces are the Vøring- and Møre Basin offshore Norway
identified by Skogseid et. al. in 1982 [11] and further discussed by Svensen
et. al. 2004 [9] and Planke et. al. 2005 [8], the Tunguska Basin in Siberia
[12] and Karoo Basin in South Africa [13]. The magmatic intrusion causes
heating and thus boiling of water and rapid maturation of organic material
in aureoles within the sedimentary basin [5]. Evidence of high fluid pressures
in the sediments around an intrusive body can be seen in figure 2.1.
When conditions are right, i.e. when the processes causing the pressure
build up is quicker than the processes of pressure relaxation, these processes
may lead to phreatic volcanic activity by breaking the pore space and localize
the flow through the overlaying sediments. A discussion of this will be given
in section 2.3.
3
Sills are tabular igneous intrusions that are dominantly layer parallel with diameters
up to 20 km. Many sills have transgressive segments that crosscut the stratigraphy. That
is why one has introduced the term saucer shaped sills.
9
CHAPTER 2. GEOLOGICAL BACKGROUND
Figure 2.1: In this figure the dolerite is intruded by fluidized previously consolidated sediments interpreted as evidence of high fluid pressure gradients
within the sedimentary rock. The hot dolerite (magmatic intrusive equivalent to basalt) intrudes the host sedimentary rock causing boiling and rapid
maturation in the aureoles. The picture is taken by I. Aarnes of a sill roof in
Golden Valley, South Africa. This is also confirmed by previous field studies
e.g. [14], [15] and [16].
The htvc are today seen as evidences of localized flow as cylindrical conduits that pierce the sedimentary strata. The piercement structures of hydrothermal vent complexes have been described as pipe-like structures formed by
rapid, localized transport of water and hydrothermal fluids onto the paleosurface. A schematic interpretation of the geological setting of hydrothermal
vent complexes can be seen in figure 2.2.
The htvc represent rapid pathways for gas produced in the contact aureoles to the atmosphere. When the gases are expelled quickly enough, it
would potentially induce global climate changes [9]. Measurements of the
maturation of organic material (shown as vitrinite reflectivity, %Ro) in contact aureoles around sills have been performed. E.g. in Brekke 2000 [17] they
measured the abundance of organic material in the aureoles found that vast
amounts of organic material lacked from the sediments.
Dickens et. al. [18] proposed the global climate to heat about 5-10
degrees leading to significant changes in the palaeontology record marking
the transition between the paleocene and Eocene epoch ('55 mya). This
coincides in time with the timing of the formation of htvc that reached the
paleosurface offshore Norway in the Vøring and Møre basins. The Paleocene
10
2.1. HYDROTHERMAL VENT COMPLEXES
A
B
C
Figure 2.2: Figure A shows a schematic interpretation of how volcanic intrusion is linked to the formation of hydrothermal vent complexes. In the
seismic image in B and C are the high amplitude reflections interpreted as
sill intrusions. Between the sill intrusion and the eye structure at the paleosurface a disrupted zone interpreted as being the piercement structure
(i.e. hydrothermal vent complex) is observed. Boiling and rapid maturation
of the organic compounds in the aureole around the sill intrusion increase
the fluid pressure that lead to the flow localization through the piercement
structure.
is characterized by the rapid expansion of mammalian stocks4 and abundance
of nummulites5 .
Similar volcanic and metamorphic processes may also explain the climatic
events associated with the Siberian traps (marking the start of the Mesozoic
era ' 250 mya) and the Karoo Igneous Province (in the Jurassic period ∼
180 mya) as well. The start of the Mesozoic era coincides with the extrusion
of 90% of the life in oceans, and 30% onland, suggesting the existence of
dramatic climatic changes world wide [20].
It is therefore suggested that intrusive magmatism, over pressure generation and venting had important effects on the climatic history of the earth.
4
Mammals such as horses, whales and bats appeared for the first time in the fossil
record in this epoch.
5
Nummulites is a genus of larger class of molluscs living in warm, shallow, marine
waters evolved early in the Eocene epoch. In some areas they are numerous enough to be
major rock formers. From Oxford Dictionary of Earth Science [19].
11
CHAPTER 2. GEOLOGICAL BACKGROUND
2.2
Kimberlites and kimberlite pipes
This presentation of the geology of kimberlites and kimberlite pipes is based
on a nice review given by Walters et. al. 2006 [6].
Kimberlites are ultramafic6 , volatile-rich volcanic rocks occuring in continental settings. Kimberlite magmas can transport trace quantities of diamonds from the deep mantle. Our knowledge today are thus mainly derived
from observations on mined kimberlite bodies.
Two principal theories have been put forward to explain the mechanisms
of kimberlite volcanism. The first is the exsolution of magmatic volatiles
(e.g. Clement and Reid 1989 [4]), and the second theory is the interaction of
rising kimberlite magma and ground water (phreatomagmatism, as proposed
by e.g. Lorenz 1985 [21]). Both of these theories have in common that the
physical process inside the structure of a kimberlite pipe is driven by high
fluid pressure gradients and the interaction between gas and varying degrees
of consolidation of granular media. Thus gas flow induced fluidization has
been invoked to explain the structure and geometry of kimberlite pipes [3],
[4] and [6].
Three types of kimberlite bodies have been recognized [6]. Each type is
characterized by their geometry and different geology. They have in common
the pipe like structure; a trace of that fluidization is the key forming process.
Class 1 kimberlite bodies are found in hard crystalline basement rocks (e.g.
the Kimberley and Venetia kimberlites in Soth Africa). They consist of steepsided carrot-shaped pipes, comprised of three distinct zones, the root zone,
the pipe zone and the crater zone. The crater zone is rarely preserved due
to post-emplacement erosion. They are hypothesized to extend as deep as
∼ 2 km and can have diameters up to several hundreds of meters [21]. Pipe
walls dip inwards at 75-85o [4].
Class 2 kimberlite bodies are thought to comprise wide (< 1300 m) and
shallow (< 200 m) craters that are filled predominantly with pyroclastic7
material (e.g. Fort à la Corne Kimberlites in Canada). These kimberlites are
emplaced through poorly consolidated sediments.
Class 3 kimberlite bodies are small steep-sided pipes which are filled
with re-sedimented volcaniclastic kimberlite (e.g. Lac de Gras kimberlite
in Cananda). Hypabyssal8 kimberlite rocks have been found in some of these
6
Ultramafic rocks are igneous rocks with low silica content. The mantle is another
example of a ultramafic rock.
7
Pyroclastic rocks (“tuffs”) consists of fragmented products deposited directly by explosive volcanic eruptions. The pyroclasts are not cemented together. The word is derived
from Greek where ’pyr’ means fire and ’klastos’ means fragmented.
8
Hypabyssal is a term for rocks that has solidified within minor intrusions, especially
12
2.3. CONDITIONS FOR VENTING
bodies. The class 3 kimberlites are found in settings where the basement
rocks are covered by a layer of poorly consolidated sediments. These are
shallow kimberlites that extend to depths down to 400 to 500 m into the
basement.
Walters et. al. 2006 [6] concludes in their paper that the formation of the
different types of kimberlites are determined by the geology of the basement.
Figure 2.3: This picture shows the core idea of the structure of a Kimberlite
pipe. The stamp was issued by Lesotho in 1973. It has become famous for
its misspelling of “Kimberlite” (from www.iomoon.com/kimberlite.jpg). The
structure of the fluidized zone is similar to what we observe in the laboratory
when fluidization is the key physical process.
2.3
Conditions for venting
Jamtveit et. al. 2004 [5] discusses under which conditions flow localization
might occur in nature. Their presentation is based on boiling of water as the
cause of the pressure build up in the aureoles around the sill intrusion. By
introducing the dimensionless venting number V e defined by the difference
between the fluid Pfmax
luid and hydrostatic pressure Phyd normalized by the
as a dike or sill before reaching the earth’s surface.
13
CHAPTER 2. GEOLOGICAL BACKGROUND
hydrostatic pressure,
Ve ≡
Pfmax
luid − Phyd
,
Phyd
(2.2)
they discuss under which conditions venting might occur. Substituting in the
maximum fluid pressure due to boiling of water given by the relative rate of
heat transport (pressure build up) and porous flow fluid transport (pressure
decay),
∆ρboil
Ve ' 2
ρβPhyd
1
=
βPhyd
s
s
κT
κf luid
κT
κf luid
(2.3)
(2.4)
where it is assumed that ∆ρboil ' ρ/2.
By using that κT is the heat diffusivity and κf luid = βφµκf luid is the hydraulic
diffusivity where β' 10−8 Pa−1 is the fluid and pore compressibility, φ is the
porosity, κ is the permeability, and µf luid is the fluid viscosity, the V e number
can be rewritten as,
1
Ve '
107 Z
10−7
√
'
Z κ
s
µf luid κT
κβ
(2.5)
(2.6)
where Z is the intrusion depth in km.
If V e 1 the sill is emplaced in an environment that is sufficiently
permeable to prevent significant fluid pressure build up since the pressure
diffusion is more rapid than the rate of pressure production. For shallow
emplacement depths in low permeable sediments, V e 1, they expects the
fluid pressure to increase and get a “blow out” situation when the fluid pressure exceeds the lithostatic pressure. In granular materials the fluidization
criteria is defined in a similar way; fluidization occurs when the fluid pressure
is larger than the weight of the overburden [22].
The above explanation of when to expect venting to occur is based on
water as the driving fluid. From thermodynamic we know that the critical
point 9 of water, is at Tc ∼ 647K and Pc ∼ 22 MPa. Hence at lithostatic
pressures exceeding 22MPa there is no discontinous jump in density and
the expression for the venting number is potentially flawed. This yields the
existence of a critical depth Z ' Pc /ρg ' 1.1 km driven by boiling of water.
9
The critical point of water is given when water in liquid and gaseous form is indistinguishable and no rapid phase change occurs
14
2.3. CONDITIONS FOR VENTING
Figure 2.4: Seismic profile of a htvc with its top located at 1.75 s (twt),
in the Eocene deposits above the terminations of high- amplitude reflections
interpreted as sill intrusions at ca 5.5 s. Between the eye structure and the sill
intrusions the zone of disrupted data interpreted as representing the conduit.
The estimated height of the piercement structure is ∼ 4 km. Seismic picture
interpreted by B. Mattson (unpublished).
Hydrothermal vent complexes are commonly interpreted to have roots
as deep as ∼9, km e.g. the vent complexes in the Mid-Norwegian volcanic
margin that erupt at the Eocene paleosurface (figure 2.4). Hence the model
for vent formation as presented by Jamtveit et. al. in 2004 is insufficient to
explain the formation of this set of vents. Though boiling of saline solutions
increases the critical point. A second model proposed to explain the deeply
rooted vents, are pressure build up mechanisms due to rapid maturation and
gas production in organic rich sediments. In combination with a permeability increase, the deeply rooted hydrothermal vent complexes might still be
explained using the V e-number. Thus when a vertical fracture opends the
15
CHAPTER 2. GEOLOGICAL BACKGROUND
pressure is reduced at the bottom of the fracture thus boiling of water can
happend at much deeper depths than 1.1 km. This is not been proved yet.
2.4
My experiments in this setting
Borehole data reveales the occurence of brecciation within the piercement
structures (Svensen et. al. 2006 [23]). When increasing the pore fluid pressure tensile fractures form in cohesive rocks causing brecciation (hydrofracturing) [10]. Brecciation increases the porosity thus also the permeability.
E.g. the Carman-Kozeny relation (equation 4.58) can be used to describe
the dependency of the two. The gas flow focuses through the high permeable
zones. In the htvc and kimberlites the focusing of the flow increases the
flow velocity sufficiently to fluidize the brecciated elements (granular media)
within the high permeable conduit zone.
The piercement structures (htvc and kimberlites) appear to be pipelike
structures on large scale. Similar pipelike piercement structures are also
found after forcing air through granular media [3], [6]. The similarities suggests that they are formed by the same physical process, namely fluidization.
Studying the fluidization or liquifaction of granular media on the laboratory
scale may be applicable to geological settings where the granular media are
the brecciation within the fluidized zone.
According to e.g. [6] the processes of fluidization of granular media is a
poorly understood. It is of major concern to understand and calculate the
phase diagram of granular meida [1], [2] to identify under which conditions
flow localization and venting occurs. Two hypothesises will be presented to
explain the onset of fluidization in granular meida:
Hypothesis 1 is that fluidization occurs when the fluid pressure at depth
Pf equals the lithostatic pressure Fg /A of the overlaying sediments. By using Darcy’s law relating the pressure and velocity we can obtain estimates
of how the flow velocity scales with depth which can be compared to the
experimental results. This hypothesis is used to determine the onset of fluidization and formation of the vent structures in several papers on htvc, e.g.
[5], [8], and [10]. It is acknowledged that the fluid pressure needed to fracture
the rock might increase with depth. But this effect is related to the process
of opening tensile fractures, not fluidization specifically. This fluidization
criteria will be tested in the thesis.
The second hypothesis on fluidization of granular media related to balancing viscous drag and gravity, FD = Fg . The granular media behaves as
a liquid when the viscous drag on each grain equals the gravitational force.
This criteria of gas-fluidization is well established within enginering and geo16
2.4. MY EXPERIMENTS IN THIS SETTING
logical systems, e.g. are Freundt et. al. in 1998 [22], Abanades et. al. in
2001 [24], and Kunii and Levenspiel in 1969 [25].
In order to test the two hypothesis of fluidization against each other and
determine the onset of fluidization, an experimental setup was build during
2003-2004. For several reasons did many of these experiments fail to succeed.
This thesis is a follow up of the previous work and a series of experiments
were performed during 2005-2006 to identify the onset of fluidization.
In these experiments air was injected into a bed of glass beads and by
increasing the flow velocity we investigate the transition between Darcy flow
and fluidization of and flow localization through the granular media. A
presentation of the experimental study with results, discussion and application to the natural processes is given in chapter 5 and 7.
17
CHAPTER 2. GEOLOGICAL BACKGROUND
18
Part II
Theoretical background
19
Chapter 3
Granular media
In this chapter a brief introduction into some properties of granular materials
relevant to the experimental study will be given. Unless other references are
given, the presentation in this section is based on two similar overviews, see
[26] and [27].
A granular media is defined to consist of macroscopic, solid and discrete
particles with a gravitational energy mgd much larger than kb T . Thermal
fluctuations is thus of negliable importance. The lower size limits for grains
in granular materials is about 1 µm. On the upper size limit, the physics of
granular materials may be applied to ice floes, where individual grains are
ice bergs. Others such as in Jaeger et. al. 1996 [27], the inelastic collisions of
granular media in the gas phase has can explain the clustering of very large
structures in density maps of the visible universe where the individual grains
are made up of planets. Thus the physics of granular materials spans a wide
variety of phenomena with many possible applications.
Most commonly used examples of granular media are flour, rice grains,
gravel, and sand. At grain level the physics is purely classic involving contact forces, gravity forces, and motion. The grain-grain contact forces in
granular media are pure repulsive, cohesive, and frictional. Taking this into
consideration, one might think that the physics of granular media is fairly
simple. But even though the physics on grain-grain scale is well understood
and within the classical domain, the bulk properties of the granular media
is not. In fact granular media has been studied for at least 200 years1 , there
are still several aspects, such as how the packing history is relevant to compaction and stress patterns, that are not yet fully understood. Further when
attempting a hydrodynamic approach to granular flow, we are still at loss as
1
Notable old names are such as Coloumb who in 1773 introduced the ideas of static
friction, Faraday who in 1831 discovered convective instability of vibrated powders, and
Reynolds who in 1885 introduced the concept of dilatancy.
21
CHAPTER 3. GRANULAR MEDIA
to how to treat the boundaries correctly since it is obvious that the nonslip
boundary conditions are invalid.
Bridging what happens on particle level to what is observed on larger/human scales is of major importance2 and raises new fundamental challenges to
physicists. Due to the fact that mga kb T and the interaction between the
particles are dissipative, the normal thermodynamic arguments break down.
Thus the phase space of granular media is independent of temperature.
The physics of granular material plays an important role in many geological processes, such as river formation, land slides, erosion, earthquakes,
and even plate tectonics that determines much of the morphology of the
earth [27]. An attempt of expanding the geological relevancy of the model
of granular media is given in this thesis by explaining the formation of kimberlites and htvc in the model of granular media that are pipe like piercement
structure formed by air-flow-induced fluidization and flow localization.
Commonly granular media is divided into three different states; solids, liquids and gases. The first two states will be presented in the coming sections.
The gas phase is not considered relevant for this study.
3.1
Granular solids
The solid state is recognized by being at rest and exhibits several interesting
phenomena, such as force networks and Janssen wall effect (see respectively
section 3.2 and 3.2.1), that separates it from ordinary solids. Another example of a peculiar property of granular media was pointed out by O. Reynolds. In his paper from 1855 [28] he identified that a compacted granular
media has to increase its overall volume to undertake any shear deformation.
The effect is called dilatancy. The classical example of this process is illustrated by walking on the beach. When we place the foot on the wet sand,
we shear, thus increase the porosity of the bulk under the foot. The water
flows from the surface into the bulk and the sand around the foot dries. O.
Reynolds explained this as a geometrical property.
Other examples of peculiar properties of static granular matter are the
large fluctuations in the pacing density, the angle of repose, and the dissipative nature of the inter particle forces.
So even though the physics on grain-grain level is fairly well understood,
several puzzeling bulk properties exists. The standard averaging procedures
developed within Statistical Mechanics does not seem to apply to understand
the transition between micro- and macro scale. One might therefore say that,
the bulk exhibits quantities that isn’t recognized in a sum of the units[29] .
2
E.g. pharmacy and oil industry.
22
3.1. GRANULAR SOLIDS
3.1.1
Packing of static granular media
The packing densities of frictional granular media under low pressure span a
wide range from the random close packing (RCP) to the random loose packing
(RLP) [30] and [31]. A thoroughly discussion of the packing of granular media
can be read in J. Feders book of “Liquid flow through granular media” [32].
This section will be limited to giving a presentation of the three terms in the
first sentence: packing density, random close packing (RCP), and random
loose pacing (RLP).
The packing density c is defined through,
c=
V olume occupied by grains
.
Sample volume
(3.1)
Along the same line can the porosity φ be defined as, φ = 1−c. It is assumed
that the packing density of spheres can vary between two well defined limits,
the RCP and RLP, dependent on the packing history. The RCP limit is
defined by the highest possible random packing of mono disperse spheres
when neglecting boundary effects. This limit can be obtained by gentle
shaking and tapping the sample.
In contrast, the RLP limit is defined by the lowest packing density that is
still mechanically stable under external load. In sand piles the lowest external
load is only the weight of the particles.
Numerous experimental studies through the last decades has investigated
these limits, some of the results and references are given below,
cRCP = 0.6355 ± 0.05 by [33], [34] and
cRLP = 0.555 ± 0.05 by [30].
(3.2)
(3.3)
When pouring spheres into a container the packing density will be somewhat
less [35] than the close packing. This is supported by the measurements in
the experiments presented in this thesis where it is found that c = 0.615.
In contrast the maximal packing density by considering the face centered
cubic lattice (FCC) or hexagonal close packing (HCP), are well known concepts from solid state physics. When stacking two dimensional layers on
top of each other the maximal packing is obtained both for FCC and HCP
packing. The packing density of this packing is
c=
π
16/3πa3
√
√ = 0.7404...,
=
16 2r 3
3 2
(3.4)
√
where a is the sphere radius, and the cubic unit cell has the volume (2 2a)3 ,
and there are four spheres in each cell with individual volume 4πa3 /3.
23
CHAPTER 3. GRANULAR MEDIA
As the packing density of granular media is reduced, obviously, the average distance between the particles is increased. This effect can be described
by a density-density correlation function [32] G(r). The function basically
answers the question; given a particle in the origin, what is the probability
to find a particle at the position r? G(r) can be found by optical diffraction
measurements of scattered light through a granular packing. This yields a
measure of the structure factor S(r). G(r) can now be found by taking the
inverse Fourier transform of S(r). When the distance between the particles
within a bed is increased the number of “paths” transmitting the force is reduced. So the packing of spheres in granular media effects the force networks
within the bed (see a discussion of force networks in section 3.2).
3.1.2
The angle of repose
The natural stable surface inclination, the angle of repose θr , of dry granular
materials is well known effect [36]. When the angle is lower than a maximum
angle, θmax the surface is stable even if the surface stress is nonzero. θmax
is defined to be the angle causing surface avalanches stabilizing the pile at a
lower angle. At angles θr < θ < θmax the pile will sometimes flow dependent
of the preparation history of the sandpile.
The definition of the angle of repose is ambiguous in the sense that the
critical angle is well defined and measurable two separate ways. When a thin
cylinder half filled with grains is rotated slowly, with the axis of symmetry
in the horizontal direction, the material is carried along with the motion of
the drum until the maximal angle of stability θmax is reached. At this angle
the grains starts avalanching and thus rapidly lower the angle. Then the
bed settles, and when rotated up to θmax again and a new avalanche occurs.
With higher rotational speeds, the avalanche frequency increases, until a
continuous surface of flow of particles occur with a well defined dynamical
angle of repose.
When granular media is poured onto a flat plate in a cone-like manner,
the inner angle of the cone defines the static angle of repose. The angle of
repose depends on factors such as size, density, surface roughness of both the
grains and the plate, and cohesion (induced by moisture[37]) [38].
3.1.3
Inter particular forces in granular media
This section will start by identifying the forces on grain-grain level of a granular packing. As previously described, the forces within a packing of granular
media are classical in the sense that the temperature plays no significant role.
24
3.1. GRANULAR SOLIDS
On the grain-level there are basically three forces acting; the radially repulsive forces, the radially cohesive forces, and the transverse frictional forces
when grains slide onto each other.
Repulsive forces
The nature of the repulsive forces between grains in granular media is material dependent. For solid spheres the repulsive forces are short ranged
and purely elastic, given by a Youngs modulus times the displacement. E.g.
spherical glass beads have a measured Youngs modulus of ∼ 72GPa [39].
Historically, the study of the complex behavior in granular systems has been
done in the elastic regime (e.g. [40]).
Other rheologies might also be taken into consideration. When subjected
to load the spheres might deform and compact plastically, as in nature during
slow compaction of sediments [41]. Not many studies have been performed
on plastic granular spheres, though Uri et. al. performed an experimental
study in 2005 [42]. They found that the radial distribution function G(r)
is reduced vertically, broadened horizontally, and shifted compared to what
is observed for hard spheres. They conclude that the rheology of the single
granular particle have to be taken into consideration when investigating the
compaction history of a granular package.
Other rheologies such as viscous and ductile might also be interesting
when considering the repulsive forces within a granular packing. The rheology feed back on the single beads has severe impact on bulk properties such
as packing density and porosity. The non-elastic rheologies is an examples
of the dissipative repulsive forces in granular packing.
Cohesion
Attractive cohesive forces on grain level has severe effects the bulk behavior in granular media. Granular materials with cohesion (often “wet” or
cemented), differs significantly in their properties from “dry” granular media.
Experimentally it is found that by adding liquids or applying a homogeneous
magnetic field one can induce attractive inter particle forces in a granular
packing. The liquid will for spherical beads settle in liquid bridges between
the grains and induce a adhesive force proportional to the surface tension of
the liquid and the size of the bridge [37].
Several studies on how the cohesive forces effects the angle of repose has
been performed. Forsyth et. al. 2001 [38] concludes that both the dynamical
and static angle of repose were found to increase approximately linearly when
increasing the inter particle forces. In these experiments the cohesion was
25
CHAPTER 3. GRANULAR MEDIA
induced by a magnetic field. These results can be in conflict with Halsey et.
al. 1998 [37]. In this paper they do theoretical stability analysis of humid
sandpiles, and find that the critical angle defining the stability of sandpiles
are unchanged when increasing the adhesive forces (humidity) for infinitely
large systems. On the other hand they report that an increase of this angle
for finite-sized systems, which is the case in [38]. Tegzes et. al. 2002 [43]
and Fraysse et. al. 1999 [44] studied how the critical angle depended liquid
content and vapor pressure respectively by using the rotating drum method.
They both found a positive relation for the critical angle. Together with
the results from the famous paper “What keeps sand castles standing?”(the
answer was obviously water) by Hornbaker et. al. in 1997 [45], it might be
concluded a positive dependence between the angle of repose and the liquid
content of the bed.
The flow characteristics of granular media is also suggested to change
when adhesive forces are induced. Experimental studies such as [43] and [46]
supports this when a transition between free flowing and stick slip behavior
is reported to happen at a critical ratio of the inter particle force and weight
of the bed.
Also the packing density of spherical granular media is dependent of the
cohesive forces in granular material. Forsyth et. al. 2001 [47] performed
studies of the packing density with their experimental setup when varying
the cohesion by inducing a homogeneous magnetic field. They conclude that
the packing density is determined by the ratio of the inter particle forces
to particle weight, regardless of particle size above a ∼25 µm. They claim
that this effect is not limited by magnetic systems, but that it shows to be
a universal effect. The cohesion reduces the particles ability to minimize its
local potential energy and relax into the lowest local position, thus increasing
the pore space.
Fluidizing vs fracturing
Another very interesting behavioral transitions due to the cohesion can be
studied when increasing the pore pressure within a granular packing. In cohesion less “dry” cases, the bed fluidizes when the viscous drag from the induced
flow supports the weight from the overlaying sediments. Fluidization occurs
when the the granular media flows or exhibit liquid-like properties such as a
drastically reduction of the angle of repose. More of the fluidization transition
in granular media in section 4.10. This was now at low cohesion. At high
cohesive forces, an increase of fluid pressure opens up tensile fractures along
the direction of largest direction of stress. From classical failure envelope
discussions, see section 3.3.1, this is as suspected. By varying the cohesive
26
3.2. FORCE NETWORKS
forces it has been proposed to experimentally investigate the competition
between fluidization and fracturing.
Friction
Frictional forces enters the play when grains slide onto each other. The energy
released by friction are dissipative in nature. To keep a granular material in
a liquid state, it continuously needs input energy (power). The friction also
severely effects the packing of granular bed, denying the particles to minimize
their local potential energy by settling between its neighbors.
3.2
Force networks
Inter particular forces in granular media forms inhomogeneous force distributions [48]. Experiments [49] has revealed that at some situations a relative
small portion of the particles bear most of the weight inside a sand pile. This
effect is often termed “arching”, which is in principle the same as the force
distribution in old stone bridges.
By laying carbon paper at the bottom of a granular packing the normal
forces on individual beads have been measured by Liu et. al. [50]. Relating
the spots left on the carbon paper to the normal force acting on the bead
enables one to calculate the normal force probability distribution. For large
forces, the distribution function was found to be,
P (σv ) = M e−βσv ,
(3.5)
where σv is the normal force acting at the bottom, M and β are experimentally determined constants (β has units N−1 .). The concept of force networks
and various models (e.g. the q-model [50]) explaining this behavior is in
detail discussed by Løvoll 1998 [26].
Arching formation is important in understanding many properties of granular materials. Janssen’s wall effect can be understood by considering the
case when the arches end at the wall of the container. Then the frictional
force between the beads and the wall would bear a portion of the weight of
the bed. The frictional force is given by the horizontal component of the
force arch times the frictional coefficient between the beads and the vertical
wall.
3.2.1
The Janssen law of wall effects
For dry cohesion less granular media, frictional wall effects reduces the normal
stress in the bottom of a silo, hopper, or container. In 1895 Janssen derived a
27
CHAPTER 3. GRANULAR MEDIA
simple mechanical model taking this effect into consideration [51]. Mourgues
2003 et. al. [52] gave an nice presentation of the concept which I will use in
this section.
Janssen assumed that the horizontal stress σh in a container was linearly
proportional to the vertical stress σv through,
σh = K j σv ,
(3.6)
where Kj is a lateral stress ratio which depends of the granular material. For
a close packing of spheres, Kj = 0.58 [53]. When considering a cylindrical
container filled with sand the weight of the bed equals the vertical stress
plus the horizontal stress times the frictional coefficient (i.e. the frictional
force) derived from Janssen’ assumption. Balancing forces over a horizontal
element dz, yields
Aj dσv + Kµw P σv dz = ρgAj ,
(3.7)
where Aj is the cross sectional area of the sand, P its perimeter, µw is
the sidewall frictional coefficient, and ρ is the sand density. By integrating
equation 3.7 over z and using that vertical stress σv is zero at the surface,
yields
ρgD
σv =
(1 − exp(−4Kj µw z/D)) ,
(3.8)
4Kj µw
where we have introduced D as the diameter of the cylinder. In [54] D.
Gidaspow identifies the coefficient to be Kj = (1−sin θif )/(1+sin θif ), where
θif is the angle of internal friction as given by the constituent equation for
powder and granular media (see section 3.3). A first order Taylor expansion
for small z shows that the vertical stress is linearly dependent of z, with
the slope ρg which reproduces the hydrostatic formula for fluids. For larger
fill heights, or small vessel diameters, we see that the vertical stress tends
ρgD
asymptotically to a constant value given by σvasym = 4K
, found by letting
j µw
z → ∞. From a large enough depth zj , adding sand does not increase the
vertical stress at the bottom of the container, i.e. the weight is carried by
the wall due to sidewall friction.
Measuring the cohesion
An example of where the Janssen effect should be taken into consideration,
is when measuring the cohesive forces in granular media. The normal way
of finding the cohesion within a granular media, is to plot measurements
of the shear stress against the normal stress and interpolate down to zero
normal stress. Then the cohesion is obtained by reading off the value from
the vertical shear axis. In [55] W.P. Schellart used this method to investigate
28
3.3. MOHR CIRCLES
the cohesion within glass micro spheres with grain sizes ∈ [400, 600] µm. He
found the interpolated cohesion to be C 0 = 137 Pa for large fill heights, when
normal stresses were larger than 600 Pa. To complement the studies he did
a series of shear tests for normal stresses of 50-900 Pa by using a smaller
cylinder. By doing so he obtained a failure envelope containing two parts;
a downward curved part near the origin (for normal stresses smaller than a
critical value of 250-400 Pa) and a linear part for larger normal stresses.
Without taking the wall effects into consideration, he interpreted the
cohesion to be given by an interpolation from the high normal stresses (given
by the fill height) only. He thus might have over-estimated the cohesive forces
within the bed, and under-estimated the frictional coefficient3 . This due to
the fact that he over-estimated the normal stress that acted on the failure
surface, by neglecting the fact that some of the weight could hang on the
wall. By correcting Schellarts data by using Janssen’s model of wall effect
Morgues et. al. 2003 in [52] found the cohesion to be reduced to C = 66 Pa
and the coefficient of internal friction to be µ = 1.6.
One might suggest, as a result of the former discussion, that the energy
necessary to fluidize the bed will increase in a non-linear way due to the wall
effect. This question will be discussed in the chapter 5.
Disorder (due to several of the presented phenomena) and strong history
dependence, makes granular systems hard to investigate and induces large
fluctuations in the measurements, thus reducing the reproducibility of the
experimental results.
3.3
Mohr circles
A convenient way of visualizing the stress state is the the Mohr diagram 4 ,
where the shear σs and normal stress σn are plotted against each other. For
a given stress in a point, the normal stress and the shear stress components
for planes of all possible orientations plot onto a circle called the Mohr circle.
The maximal and minimal stress components, have their values defined by
the intersection of Mohr circle with the σn axis, thus defining the principal
stress directions. The radius of the Mohr circle is defined by half the diameter
which is given by the difference between the absolute value of the maximal
min |
and minimal stress components, Rm = |σmax |−|σ
.
2
3
The frictional coefficient is given was given by C. A. Coulomb to be µ = tan θ, where
θ is the slope in the shear versus normal stress measurements. To be revisited in section
3.3.2.
4
Christian Otto Mohr (1835 - 1918) was a German civil engineer. He is most famous
for his contribution to the theories of mechanics and strength of materials.
29
CHAPTER 3. GRANULAR MEDIA
Now the orientation of a physical plane is defined by how its normal n
is oriented relative to a known coordinate axis. Since the measured angle
of the physical plane takes values from 0o to 180o , the angles in the Mohr
diagram are doubled. The normal and shear stress components that acts an
a given plane with an angle θM plots (σs and σn ) at the end of the radius
with an angle 2θM in the Mohr diagram. The stress components that lie at
opposite ends of any diameter on the Mohr circle are the components that
act on perpendicular planes in physical space.
The conjugate planes of maximum shear stress are the planes whose normal lie ±45o off of the maximum principal stress direction in physical space.
In the Mohr diagram these plots on the top of the circle, at ±90o , thus
defining the maximum shear stress.
Now the magnitude of the stress at a point is uniquely characterized by
two scalar invariants of the stress. The first of the two are located in the
center of the Mohr circle, and are thus given to be the mean normal stress,
thus the fluid pressure is given by,
p=
σmax + σmin
.
2
(3.9)
The other scalar invariant is the radius of the circle as previously defined.
These two values p and Rm , are called scalar invariants because they are
scalars whose values are the same for any set of components that define the
same stress. Thus by knowing these two variables we are able to completely
construct the circle.
3.3.1
Failure envelopes/constituent equations
When a solid breaks, several types of fractures are observed. Armed with the
Mohr diagram, several of the observed features can be described (see figure
3.1) where we have plotted how different failure envelopes and fractures are
related. The Mohr diagram consists of three main parts (figure 3.3.1), that
will be presented in the coming sections.
3.3.2
Coloumb fracture criterion
Coloumb5 wrote his first paper in 1773 considering a number of problems
involving the strength of materials such as wood, stone, and soil. He observed
that the strength of the materials could be derived from two sources: cohesion
and friction. Within soils he observed that failure usually were associated
5
Charles Augustin de Coulomb (1736 - 1806), was a french military engineer who worked
on applied mechanics but he is best known for his work on electricity and magnetism
30
3.3. MOHR CIRCLES
Figure 3.1: Failure envelopes and related fractures. We see that the Mohr
diagram mainly consists of three parts; the Griffiths, Coloumb and Von Mises
ductile failure criterion. The Griffiths failure criterion (parabolic failure criterion) is relevant for high cohesive rocks. When cohesion is larger than
the differential stress, a mode 1 tensional fracture forms when fluid pressure
within the rock is increased. The Coloumb criteria can be reached by increasing the fluid pressure, when the differential stress is larger than the cohesion
within the bed forming Mode 2 shear fractures along the internal angle of
friction. The Von-Mises ductile failure criterion is relevant for ductile materials and will not be discussed in this thesis. Figure is taken from Twiss and
Moores book on Structural Geology 1992 [56].
with a surface within the soil. Restricting attention to this failure surface,
he wrote his failure criterion as
σs = C + σn tan θ,
(3.10)
where he proposed that the shear stress σs was given by the cohesive force
C within the soil plus the tangent of the angle between the failure plane and
the maximal stress direction times the normal force σn tan θ. The cohesion
has dimension of stress [Pa]. This angle is later in idealized situations been
identified as the internal angle of friction θ = θif , as has been described
earlier.
The Coloumb failure criterion is a straight line in figure 3.3.1, forming so
called Mode 2 fractures. The Coloumb failure criterion is still used today for
several applications, hereby also in granular media.
31
CHAPTER 3. GRANULAR MEDIA
3.3.3
Tensile fracture criterion
Within high cohesive materials, tensile fractures are observed when fluid
pressure is increased. The tensile or Mode 1 fractures, propagate in the
largest stress direction. This effect can be used to determine the major
stress direction within the media of interest. In porous media the largest
direction of stress can be altered by horizontal extension or compression.
If the cohesion is relatively higher than the differential stress, the fractures
will propagate vertically and horizontally respectively if the maximal stress
direction is altered by the extension and compression.
3.3.4
Von-Mises failure
The von Mises6 failure criterion is applicable to ductile materials e.g. metals.
R. von Mises suggested in 1913 that yield will occur when the value of the
shear stress reaches a critical value irrespectively of the normal stress. This
can be written as the von Mises failure criterion
σs = σvM ,
(3.11)
where σvM is the yields stress for the material of interest. When a ductile
metal yields on a macroscopic level, the displacements occur between atoms
that make up the crystal lattice. These atomic displacements are termed
dislocations. A dislocation can move through the lattice, displace one atom
after another producing small irrecoverable deformations [57]. The materials
of interest in this thesis are granular materials, far from being ductile. A
discussion of von Mises failure and dislocations is therefore not relevant. I
therefore just mention it, and stop the discussion here. :-)
Considering the case when the radius of the Mohr circle is larger than the
cohesion within the soil, Rm > C (equivalently; when the differential stress is
larger than the cohesion). The presence of pore fluid pressure within the soil
reduces the confining pressure p within the material, defined by the center of
the Mohr circle. Thus a shift of the center of the Mohr toward lower normal
stresses is occurs by an amount equal to the fluid pressure. The radius Rm
of the Mohr circle is unaffected and when the Mohr circle touches the failure
envelope, a shear (Mode 2) fracture forms.
Vice versa, when Rm < C an increase of pore fluid pressure does not
make the Mohr circle touch the Coloumb failure criterion but the Griffiths
6
Richard von Mises (1883 - 1953) was an Austrian scientist working on, as he put it in
his own words shortly before his death, “practical analysis, integral and differential equations, mechanics, hydrodynamics and aerodynamics, constructive geometry, probability
calculus, statistics and philosophy”(E. Mach).
32
3.4. GRANULAR LIQUIDS
criteria in stead. The discussion of how the porous media in my experiments
are fractured is a matter of cohesion and stress direction. More of this in the
discussion chapter.
3.4
Granular liquids
Granular beds at rest are frequently encountered in our everyday life. Piles in
open spaces or held by boundaries, their stay motionless due to the vanishing
ratio of kB T /ρgd. However, when an external force or sufficiently amounts of
power is applied, suprising dynamics are observed not seen in other phases of
matter. Phenomena such as compaction, convection, segregation, jamming,
avalanches, pattern formation, relaxation of topography, and avalanches is
observed when granular media liquiefy.
Examples of external forces that can liquefy, or fluidize, granular solids
is to mechanically vibrate a container of grains or force liquids through the
bed. Examples of fluidization experiments of granular media are Huerta et.
al. 2005 [58] and Valverde et. al. [59] respectively. In the latter case, the
energy is transmitted to the bed by viscous drag between the fluid and the
grains. When energy is injected to these systems a kinetic energy gradient
develops through the bed due to the dissipation effects inside the bed [60].
To define the limits, within which granular dynamics can be described
by use of well established kinetic and hydrodynamic theories, is of major
interest now days, see [1] and [2]. However, models for granular flow do
not have the stature of the Navier-Stokes equations. It is well established
that the continuum equation is not defined for small volumes comparable
to the particle, or pore spaces. In the other limit, even though the largest
systems such as corn-silos etc., the systems are far from large enough to
be called infinitely large. To even complicate the picture, as described in
section 3.2, we know that inhomogeneities due to force networks can span
hundreds of particles. The issues of V → 0, V → ∞ and force networks
raises severe problems when applying similar averaging process over length
and times scales as in the Navier Stokes equations [27].
Normally in the literature it is suggested that when the granular system
flows it has become fluidized. Geldart proposed an empirical classification
of gas-fluidized powders back in 1973 [61]. He proposed that powders7 could
be categorized according to their fluidization properties. He identified three
different categories A, B and C. However we need to know the fluidization
property for category B (coarse, dense, low cohesive particles) only in this
thesis, since our material behaves within this regime. The fluidization of
7
Powders is a substance that has been crushed into very fine grains.
33
CHAPTER 3. GRANULAR MEDIA
category B powders are described in the following way; when the particles
are supported by the drag force of a low viscous fluid, the bed expands
smoothly as the fluid velocity is increased, and above a certain fluid velocity
the fluid like regime is followed by a bubbling regime. This classification has
later been widely used [59] to identify and classify different types of powder.
However, all the initially listed phenomena might identify the onset of
granular fluidization. E.g. a recent study by Huerta et. al. 2005 [58] suggest
that dry granular can be fluidized without flowing. They found by horizontally vibrating beds showed hydrostatic properties by measuring buoyancy
forces, according to Archimedes’ principle, of light spheres.
It is also proposed that the granular media is fluidized when it cannot
support the relatively large shear forces due to the angle of repose. So when
we blow air through a pile at the angle of repose, we see that at a certain
air velocity the pile suddenly collapses and relaxes onto a smaller angle of
repose. This air velocity might mark the onset of fluid-like behavior in the
granular bed.
Due to the ambiguity of the fluidization term I hereby define the transition
from static to fluidization in my experiments to be the case when the particles
start flowing forming a piercement structure from the inlet to the surface.
This definition is supported by the fact that the bed is semi static before the
piercement phenomena occurs.
A further development of the concept of fluidization will be done in section
4.10 after introducing important concepts from fluid dynamics.
3.4.1
Segregation phenomena
Segregation phenomenon might occur in dynamic granular media when particles
of different sizes, shapes, or densities are mixed. In our daily life, this is
known as the Brazil nut effect, when we always tend to find the delicious
large nuts on the top in the nut-mixture. It is also the reason why large
stones suddenly appears in the potato field, even though I know that I picked
out the stones last year. One might ask the question, who carried the stones
into the potato field during the winter? This effect is of significant practical
and conceptual relevance for example in pharmacy, and of course for potato
farmers...
This phenomena is reported to happen in rotating drums [62], vertically
vibrated boxes [63], and in air driven systems [64]. See e.g. Tarzia et. al.
2005 [65] where they analytically investigate nature of size segregation in
vibrated granular mixtures. They find a cross over between ascending and
descending of large grains when the number of small grains exceed a critical
value.
34
3.4. GRANULAR LIQUIDS
Huerta et. al. 2004 [63] reveal that different physical phenomena occurs
as they vary the frequency of the vibrations. For low frequency they find that
convection dominates when the relative density is larger than one, and inertia
dominates when the relative density is less. In contrast, in the high frequency
cases, when fluidized, the segregation is caused by buoyancy effects. A couple
of experiments shows segregation of different sized particles, see chapter 6.2.
35
CHAPTER 3. GRANULAR MEDIA
36
Chapter 4
Liquid flow in porous media
The given presentation of hydrodynamic is based on Jens Feders excellent
book, “Flow in porous media”[32]. The set of differential equations describing the motion of fluids are the Navier-Stokes equations1 (NS). This set of
equations is based on the dynamical balance of forces acting at any given
region of the fluid. Hence the changes in momentum of the particles of a
fluid are the sum of changes in pressure and dissipative viscous forces acting
inside the fluid.
It has been said that the NS set of differential equations are the most
useful set of equations in physics. They describe the physics over a large
number of phenomena of academic and economic interest. Examples where
the NS is applied is ranging from modeling the weather, water flow in a pipe,
moving stars within a galaxy, study of blood flow, air flow around a wing, to
designing cars.
Despite their indisputable importance, a deep study of the NS differential equations is out of range of an experimental master thesis in physics.
Especially the case of high Reynolds number, of turbulent flow, the study of
the NS equations gets very complicated due to the non-linear term (v · ∇v)
in the NS equation2 . What I will do in the proceeding sections is to give
a brief presentation of the basis of the NS-equation and which assumptions
that are made in its derivation. I will also look at some common simplifica1
The Navier-Stokes equations are named after Claude-Louis Navier and George Gabriel
Stokes. C. L. Navier (1785 - 1836) was a French engineer and physicist born in Dijon, died
in Paris. Sir G. G. Stokes, (1819 - 1903) was an Irish mathematician and physicist, who
at Cambridge made important contributions to fluid dynamics, optics, and mathematical
physics (including Stokes’ theorem).
2
Even though turbulence is an everyday experience, it is extremely hard to find solutions for this class of problems. A price of 1 000 000 $ was offered in May 2000 by the
Clay Mathematics Institute to whoever makes substantial progress toward a mathematical
theory which will help in the understanding of the phenomenon.
37
CHAPTER 4. LIQUID FLOW IN POROUS MEDIA
tions (Stokes flow of a sedimenting particle, Euler equation for high Reynolds
numbers, and Darcy’s law) that will be useful later in the thesis.
4.1
Derivation of the NS-equations
Newton’s second equation F = ma = dp/dt, specifies that the rate of change
in momentum p equals the exerted force. When considering a small (Lagrangian) volume V with mass m = ρV and momentum mv(r, t) the acceleration on the fluid element is not the same as on a rigid body. To discuss the
acceleration on fluid elements we must remember that the velocity field of a
Lagrangian volume element changes in both time and space, i.e. ∂v/∂t 6= 0
and ∂v/∂r 6= 0. Therefore the rate of change of momentum per unit volume
is given by the substantive derivative.
Substantial derivative
In hydrodynamics one often has to consider how a quantity changes both
as it moves to a different region and as the overall field is changing. This
effect is termed the substantial derivative. It is often also called the advective
derivative or Lagrangian derivative in fluid dynamics.
When the derivative of a field folloving the particle or the lagrangian fluid
element the substantial derivative is defined through the operator
D
∂
=
+ v∇
Dt
∂t
(4.1)
where v is the fluid element velocity, ∇ is the spatial differential operator, and
∂
is the Eulerian derivative. The Eulerian spatial derivative is the derivative
∂t
of a field with respect to a fixed position in space or time. The second term
is an advective term. This differential operator works from left on any given
vector field.
The difference between the Eulerian and susbstantive derivative is illustrated by considering steady flow3 of whater through a hosepipe with gradually decreasing cross-section. Due to mass conservation, and the fact that
water is nearly incompressible, the flow is thus faster in one end than in the
other. Since the flow is steady, the Eulerian derivative is everywhere zero
but the substantial derivative is non-zero since any individual parcel4 of fluid
accelerates as it moves down the hose.
3
∂
By steady it is meant that there is no change in time, hence ∂t
= 0.
By parcel we mean a tiny amount e.g. volume or mass. Small enough so the physical
quantity given by the field is said to be zero within the “tiny amount”.
4
38
4.1. DERIVATION OF THE NS-EQUATIONS
Thus for the flow velocity field,
ρ
Dv
∂v
=ρ
+ ρv · v.
Dt
∂t
(4.2)
Considering ideal fluids, i.e. where dissipation due to viscosity and internal friction is neglected, the force on the fluid element is given by the
pressure and conservation of momentum similar to Newton’s second law,
Z
Z
Z
d Z
ρvdV = − ρvvdS − pdS + ρgdV.
dt V
S
S
V
(4.3)
Where the terms respectively is interpreted to be:
• rate of increase of momentum of fluid in V ,
• rate of addition of momentum across a surface S by convection,
• force acting on fluid in V by pressure p, and
• force on fluid in V by gravity.
In combination with Gauss divergence theorem5 , when V → 0, we find the
Euler equation6 for an ideal fluid,
∂
ρv + ∇ · (ρvv) = −∇p + ρg.
∂t
(4.4)
By using that ∇a · b = a∇b + b∇a, we can rewrite the second term on the
left hand side as
∇ · ([ρv]v) = ρv∇v + v∇[ρv].
(4.5)
In combination with the continuity equation 7 , that
that
∂p
∇[ρv]v = ρv∇v − v .
∂t
5
∂p
∂t
+ ∇ · ρv = 0, we find
(4.6)
The Gauss divergence theorem relates the integrated flux of a vector field a across
the Gaussian surface
R S to the
R change (divergence) inside a volume V , mathematically
formulated through S adS = V ∇ · adV .
6
Leonhard Euler (1707 - 1783) was a Swiss mathematician and physicist. He is considered to be one of the greatest mathematicians of all time. Euler was the first to use
the term "function" to describe an expression involving various arguments; i.e., y = f(x).
Also he introduced lasting notation for common geometric functions such as sine, cosine,
and tangent. He found the Euler equation of an ideal fluid in 1755. A quick count on
Wikipedia tells me that about 40 topics in mathematics and physics are named in honour
of him.
7
The equation of continuity can be shown to be equivalent to mass conservation.
39
CHAPTER 4. LIQUID FLOW IN POROUS MEDIA
We can now reinsert this equation into equation 4.4 to find that,
ρ
∂
v + ρv∇v = −∇p + ρg.
∂t
(4.7)
This equation is termed the second Euler equation and is valid for ideal fluids,
where viscosity can be ignored. Now it is important to recognize that the
left hand side of the second Euler equation 4.7 is equivalent to the right hand
side of equation 4.2.
Through introducing the force density, f = −∇p + fµ + F, where −∇p
is the pressure driven force, fmu is the viscous force to be discussed later in
section 4.2, and F is a body force8 we can rewrite Newton’s equation of the
form
∂v
+ ρv∇v = −∇p + fµ + F.
(4.8)
ρ
∂t
4.2
Viscous force
To describe the viscous forces on fluid elements we introduce a stress tensor
specifying the force per unit area as illustrated in figure 4.1. The fundamental
Figure 4.1: The various component of the stress tensor.
assumption in the following is that the stress tensor σ is proportional to the
8
In physics and fluid dynamic there are two types of forces; the body force and surface
force. Body forces, such as gravity, acts on all elements of a continuum of a body which
is represented by the force acting on the center of mass. Surface forces, like stress and
friction, acts only on surface elements, whether it is a portion of the bounding surface of
the continuum or an arbitrary internal surface.
40
4.2. VISCOUS FORCE
rate of strain tensor, i.e. we have a Newtonian fluid 9 where,
σij = Λijkl ekl .
(4.9)
Assuming the fluid to be isotropic, the number of elements of the viscosity
tensor Λ reduces to three free parameters (λ, ξ, and χ) so,
Λijkl = λδij δkl + ξδik δjl + χδil δjk .
(4.10)
Rewritten with the stress tensor,
σij = λδij δkl (e11 + e22 + e33 ) + (ξ + χ)eij
= λδij ∇v + 2µeij ,
(4.11)
(4.12)
where the viscosity µ is given by the phenomenological constants ξ and χ,
µ=
1
(ξ + χ) ,
2
(4.13)
and ∇v = eii . The λ-term is the bulk viscosity and is related to the viscous
dissipation effect.
By summing up the viscous forces in x direction of a small fluid volume
element, as in figure 4.2
Figure 4.2: Viscous stresses on a volume element considering the viscous
forces in x-direction. Figure from [32].
Fµ,x = (σxx (x + dx) − σx (x))
= (σxy (y + dy) − σy (y))
= (σxz (z + dz) − σz (z)) ,
9
(4.14)
(4.15)
(4.16)
For other fluids, such as Bingham fluids, power-law fluids and incompressible fluids
other relations between the stress and strain applies.
41
CHAPTER 4. LIQUID FLOW IN POROUS MEDIA
we find the total viscous force in x-direction, Fµ,x . Now the force per unit
volume is therefore
∂σxx ∂σxy ∂σxz
+
+
.
(4.17)
fµ,x =
∂x
∂y
∂z
For the viscous forces acting in an arbitrary direction, I would kindly refer
to Jens Feders book “Flow in porous media” chapter 5 where this is nicely
done. I will here present the general answer expressing of the force per unit
volume;
fµ = µ∇2 v + (µ + λ)∇ · ∇v.
(4.18)
If we now consider incompressible fluids, that is when ∂ρ
= 0, the continuity
∂t
equation gives ∇ · v = 0. In this case the viscous force simplifies to
fµ = µ∇2 v.
(4.19)
In equation 4.8 the viscous force was a unknown variable. By using
equation 4.19 the equation of motion can be found to be
ρ
∂v
Dv
= ρ
+ ρv∇v
Dt
∂t
= −∇p + µ∇2 v + (µ + λ)∇∇ · v + F.
With the continuity equation
∂ρ
∂t
(4.20)
(4.21)
+ ρ∇v = 0, the gradient of the velocity field
∇·v =0
(4.22)
for incompressible fluids, equation 4.20 reduces to the Navier-Stokes relation
ρ
∂v
+ ρv∇v = −∇p + µ∇2 v + F.
∂t
(4.23)
This equation is a second order partial differential equation with a non-linear
term (v∇ · v) which complicates the problem of solving it. In several cases
the external force, F = 0, and the fluid movement is caused by the pressure
differences or relative movement of the boundaries only. Generally the fluid
flow takes place in a gravitational field which cannot be ignored.
When the fluid is supported in the bottom, the gravitational force is balanced by a vertical pressure gradient (uniform density). Thus the dynamical
equation can be reduced to one without body forces.
Now equation the continuity equation 4.22 is a scalar equation and NS
equation 4.23 is a vector equation, giving a total of 4 equations to solve the
scalar quantity p, and the vector quantity v. The number of equations equals
the number of unknowns thus closing the set of equations.
To solve the NS equations one needs to consider the boundary conditions
for every instant problem.
42
4.3. REYNOLDS NUMBER
4.3
Reynolds number
By introducing the kinematic viscosity ν = µ/ρ, we can rewrite the NS
equation 4.23 as,
∂v
1
+ v∇v = − ∇p + ν∇2 v,
(4.24)
∂t
ρ
for situations where the body forces is neglected. The kinematic viscosity is
thus the only material property entering the equation. For air the kinematic
viscosity is given to be νair = 0.15 cm2 /s and water νwater = 0.01cm2 /s.
For flow in a tube, or a sphere both with diameter a, the only variables
affecting the flow field are ν, a, and v. By combining this set of variables in
a dimensionless number we expect this number to characterize the different
flow regimes. The conventional choice of such a dimensionless ratio is the
Reynolds number,
va
.
(4.25)
Re ≡
ν
In Tritton et. al. [66] they investigate how the normalized pressure drop for
a given flow changes for for different Reynolds10 numbers. In figure 4.3 an
abrupt increase of pressure drop at a Re ∼ 3000 due to a drastic change in
the velocity profile can be seen. The velocity profile changes from laminar
(low Re-numbers) to turbulent (high Re-numbers) flow.
By considering the two regimes separately, we can derive the Stokes equation for laminar flow and Euler’s equation for turbulent flow, for low and high
Reynolds numbers respectively.
4.4
Euler’s equation
In the limit of very high Reynolds numbers, the viscosity term in the NSequation 4.24 can be ignored, so
1
∂v
+ v∇v = − ∇p, Re 1.
∂t
ρ
(4.26)
This equation was first obtained by L. Euler in 1755 long before the definition
of Reynolds number and is applied to cases where we have fully developed
turbulence where inertia only can be considered.
10
Osborne Reynolds (1842 -1912) was an British fluid dynamics engineer who famously
studied the conditions in which the flow of fluid in pipes transitioned from laminar to
turbulent. From these experiments came the dimensionless Reynolds number for dynamic
similarity - the ratio of inertial forces to viscous forces. A crater on Mars is named in his
honour.
43
CHAPTER 4. LIQUID FLOW IN POROUS MEDIA
Figure 4.3: In a pipe of diameter a and length L the normalized pressure drop ∆pa3 ρ/Lµ2 is investigated as a function of Re. The transition
between laminar onto turbulent flow defines the critical Reynolds number to
be Recrit ' 3000 in this case. After Tritton et. al. 1988, [66].
The drag force FD on an obstacle moving at high Reynolds number is
commonly written as,
1
(4.27)
FD = CD ρAv2 ,
2
where CD is the drag coefficient, ρ is the density of the medium, v is the flow
speed, and A is the cross-sectional area. This law is valid at low kinematic
viscosities, ν 1 (or equivalent, high Reynolds numbers Re 1), so the
resistive force is dominated by inertia.
4.5
Stokes flow and sedimentation
In the limit of low Reynolds numbers, the viscosity term dominates over
the non-linear inertial term in the NS equation 4.24. By considering the
stationary case when ∂v
= 0 and ignoring the non-linear term, we obtain
∂t
Stokes equation,
∇p − µ∇2 v = 0, Re 1.
(4.28)
This differential equation is used for creeping or laminar flow (see figure 4.5)
where the characteristic length scale l is microscopic. The drag force on a
spherical object moving in a infinite viscous fluid for low Reynolds number
44
4.5. STOKES FLOW AND SEDIMENTATION
Figure 4.4: Navier Stokes flow passed a cylinder and sphere at high Reynolds
number, Re Recrit . We see that turbulence is fully developed behind the
cylinder. The picture is taken from WWW.engineering.uiowa.edu/ cfd/gallery/images/turb4.jpg and Wikipedia respectively.
is given by Stokes law
FD = 6πµav.
(4.29)
This expression is derived in J. Feders book “Flow through porous media”,
chapter 5 [32].
Experiments done by Taneda in 1956 [67] shows that at Re ∼ 24 eddies
form behind the sphere, at Re ∼ 130 the eddies start oscillating, and at
Re ≤ 200 the flow gets increasingly turbulent. A particle moving in a viscous
fluid is equivalent to fluid flowing past a sphere. This concept is used in the
case of determining the viscous drag for a fluid flowing through a porous
medium.
Stokes flow is commonly used in the processing of polymers, and other
sedimentary settings.
45
CHAPTER 4. LIQUID FLOW IN POROUS MEDIA
Figure 4.5: Navier Stokes flow passed a cylinder at intermediate Reynolds
number, Re = 100. We see that turbulence is developing behind the cylinder.
The picture is taken from WWW.idi.ntnu.no/ zoran/NS-imgs/lics.html.
4.6
Bubble in a viscous fluid
The drag force of a spherical bubble of radius r raising slowly in a fluid of
viscosity µ and density ρ is given in Bird et. al. 1987 [68] to be
(4.30)
FD = 4πµrv
The bubble tends to be spherical when small due to surface tension and
minimization of energy. Note that the drag force is independent of surface
tension.
For finite Reynolds numbers, the inertial forces will pertubate the shape
of the bubble, and its shape will be a balance among viscous, inertial and
surface tensional forces. The shape of the bubble is shown by Taylor et. al.
(1964) [69] to be described by the radial function
R(θ) = r 1 −
5
CaRe(3cos2 θ − 1) ,
95
(4.31)
where Ca ≡ µv/γ is the capillary number11 . The expression is valid for
Re 1 and ReCa 1.
11
The capillary number is given by the relative effect between the viscous µ forces and
the surface tension γ acting across an interface between a liquid and a gas, or two nonwetting fluids. v is a characteristic velocity of the front.
46
4.7. DARCY’S LAW
Figure 4.6: Navier Stokes flow passed a cylinder at low Reynolds numbers
Re = 10. We see that turbulence is developing behind the cylinder. The
picture is taken from WWW.math.armstrong.edu/mmacalc/gallery/flow.gif.
4.7
Darcy’s law
Henry Darcy12 was the first to relate the permeability to the proportionality
between flow and pressure gradient. He did this in his paper from 1856 [70]
when he designed and executed the municipal water supply systems in Dijon,
France. His experimental setup is shown in figure 4.7. Darcy’s flow model is
widely used for all types of flow through porous media, and will also be used
in this master thesis. It is shown that the flow of oil in oil reservoirs follows
Darcy’s law.
Henry Darcy found his relation by varying the flow Q [m3 /s] through a
homogeneously porous bed and measure the head difference. He found that
Q = K 0A
h1 − h 2
,
L
(4.32)
where K 0 is a constant that depend on the type of sand that he used. The
head difference is related to the pressure difference between the inlet and
outlet through
p1 − p2 = ρgh1 − ρgL − ρgh2 .
12
(4.33)
Henry Philibert Gaspard Darcy (1803 - 1858), was a French scientist who made several
important contributions to hydraulics e.g. the Darcy’s law of flow through porous media.
He was born and lived his life in Dijon.
47
CHAPTER 4. LIQUID FLOW IN POROUS MEDIA
Figure 4.7: Darcy’s experimental setup investigating the proportionality
between the volume flux Q and pressure difference ∆p. The picture is taken
from [32].
The flux Q can now be rewritten with K = K 0 /ρg to be
p2 − p 1
Q = KA −
+ ρg .
L
(4.34)
The flow Q is zero if the pressure difference equals the hydrostatic pressure
difference, ∇p = ρgL, i.e. h1 = h2 .
In 1930 Nutting [71] introduced the permeability through K = k/µ, where
µ is the fluid viscosity, to characterize the porous medium. The dimensions
of k can be found through,
p2 − p 1
k
−
+ ρg
µ
L
[Q] · [L] · [µ]
= m2 .
⇒ [k] =
[A] · [∇p]
Q = A
(4.35)
(4.36)
Wyckoff et al [72] suggested in 1933 the unit Darcy as a measure of permeability. 1 Darcy13 is the permeability of a porous rock when the flux of 1 cm3 /s
of a fluid of viscosity 1 cP14 flows through a cross section of 1 cm2 driven by
a pressure gradient of 1 atm/cm.
13
14
In SI units 1 Darcy is equivalent to 0.9869 µm2 .
1.09 cP is the viscosity of water.
48
4.8. DARCY’S LAW ON DIFFERENTIAL FORM
4.8
Darcy’s law on differential form
In 1946 M. Muskat [73] generalized Darcy’s law to infinitesimal layers where
the volume flux per area or fluid velocity15 is defined by v = Q/A. By using
that (p1 − p2 )/L → ∇p as L → 0 and g = (0, 0, −g) we can rewrite equation
4.34 as
k
(4.37)
v = − (∇p − ρg) .
µ
The flow velocity is in the direction of the pressure gradient if the gravity term
is neglected. To fully specify the flow, Darcy’s law must be supplemented
by the continuity equation for a fluid moving in a porous medium. The
continuity constraint is given when the rate of increase of mass of the fluid
V within V equals the addition of mass across the surface S, formulated
through
Z
Z
d
φρdV = − ρvdS.
(4.38)
dt V
S
Now since the fluid is excluded from the matrix, the porosity φ must be taken
into account. When V → 0 we obtain16 the continuity equation,
∂φρ
= ∇ · (ρv) = 0.
(4.39)
∂t
Combining the continuity equation 4.39 and Darcy’s law 4.37 we obtain the
dynamic differential equation for the flow of incompressible fluids through
granular media
∂φρ
k
= −∇ ρ − (∇p − ρg)
∂t
µ
!
k
= ∇ ρ (∇p − ρg) .
µ
!!
(4.40)
(4.41)
By neglecting the gravity term, i.e. when ∇p ρg 17 and assuming constant
porosity we can rewrite equation 4.41
"
#
∂ρ
k
= ∇ ρ ∇p .
∂t
φρ
15
(4.42)
The flow velocity is also termed Darcy velocity, seepage velocity, filtration velocity, or
specific discharge. A beloved child has a lot of names.
16
The continuity equation is only valid for length scales much larger than the pore space.
By letting V → 0 is therefore controversial since the pore space has a finite size in porous
media.
17
In my experiments the measured pressure gradient across the bed is in the order of
(p1 − p2 )/h ' 104 Pa/m and ρair g = 10 Pa/m, so for my experiments I am happy to
neglect the gravity term in the dynamical equations for the fluid.
49
CHAPTER 4. LIQUID FLOW IN POROUS MEDIA
4.9
Models of permeability
In this section I will discuss two models of permeability in a porous media.
First I will examine the case of a stack of membranes with capillary holes
(the capillary model), generalized from the permeability through a single
membrane. The motivation of studying this models is that it gives important
insight to the second model; the Carman Kozeny model for permeability of
a porous media.
This section is based on the excellent presentation given by Jens Feder in
his book of “Flow through porous media” [32].
4.9.1
The capillary model
The first attempt to derive an analytical expression for the permeability of a
porous media was done through considering flow through a stack of capillary
membranes. For a single membrane neglecting end effects, the volume flux
was found experimentally by G. Hagen in 1838 [74] and J. L. M Poiseuille in
1840 [75] and later in 1845 theoretically by G. G. Stokes [76] to follow the
relation,
πr 4 ∆p
πr 4 ∂p
Q= c
= c
,
(4.43)
8µ L
8µ ∂x
where rc is the capillary radius and L is the length of the capillary tube.
The relation for the volume flux is termed Hagen-Poiseuille equation. The
average flow velocity through the capillary is given by the volume flux divided
by the area,
r 2 ∆p
Q
.
(4.44)
< u >= 2 = c
πrc
8µ L
With n pores per unit area, the total volume flux through the single membrane is,
k ∆p
πrc4 ∆p
=
.
(4.45)
v = nQ = n
8µ L
µ L
By using Darcy’s law we obtain en explicit expression of the permeability,
k=n
πrc4
.
8µ
(4.46)
Unfortunately this model is inadequate for porous media since it only
accommodates flow in one direction. Now the porosity φ of a membrane
with nA capillary tubes is given by the number of pores times the pore cross
sectional area divided by the area A of the membrane, so
φ = nπrc2 .
50
(4.47)
4.9. MODELS OF PERMEABILITY
The total flow velocity v is given by the average pore velocity < v > times
the porosity, as pointed out by Dupuit in 1836 [77]. By using equation 4.47
in 4.46 we get a relation between the macroscopic properties porosity and
permeability,
r2
(4.48)
k =φ c.
8
By introducing the concept of specific surface area S in porous beds we
can further develop our understanding of the permeability. S is defined as
the pore surface per unit volume. In the capillary tube model S is given to
be the the number of capillaries nA times their inner area 2πrc L divided by
a unit volume AL,
S = n2πrc .
(4.49)
By using equation 4.47 we can relate S to the porosity,
S = 2φ/rc .
(4.50)
Since the specific surface area S has unit [m−1 ], we expect S −1 to be a typical
pore size of the porous media. Getting an expression for the permeability
k of measurable macroscopic quantities is now easy when solving the last
expression for rc and equate it into equation 4.48,
k=
φ3
φ3
=
.
2S 2
K0 S 2
(4.51)
K0 is here introduced as the Kozeny constant, which is in this case given to
be 2. We now have developed an expression for the permeability given by
the porosity and specific surface area.
4.9.2
Carman-Kozeny model of permeability
Now as previously stated the capillary model only takes flow in one direction
into consideration. In order to develop a model for the permeability in a real
porous media, with known porosity and surface area, the non straight path of
the fluid flow has to be considered. This was first done by J. Kozeny in 1927
in [78] by considering two effects: Firstly he introduced the effective capillary
length Le and assumed the pressure drop ∆p acts over the capillary length
instead of the actual sample length. By using the Hagen-Poiseuille equation
4.43, the volume flow rate through any one of the effective capillaries can be
expressed as
πr 4 Le ∆P
Q= c
.
(4.52)
8µ L L
51
CHAPTER 4. LIQUID FLOW IN POROUS MEDIA
The total area of flow A is given by the product of the number n and
the area of capillary tubes Ac , which equals the porosity φ times the cross
sectional area of the sample As . This is summarized in,
A = As φ = nAc .
(4.53)
Secondly, he defines the number of capillary tubes given by the area of flow
divided by the area of the capillaries multiplied by the turtosity18 τ ≡ L/Le „
φAs L
πrc2 Le
= φAs L.
n =
⇒ nπrc2 Le
(4.54)
(4.55)
The last expression is just two expressions for the pore volume.
By combining equation 4.55 and 4.52 we find an expression for the flow
velocity v,
φa2 ∆p
,
(4.56)
v= 2
8τ µ L
where Q = vnAc . By using Darcy’s equation we have developed another
2
c
.
expression of the permeability k = φr
8τ 2
Since the specific surface area is given, as in equation 4.50, we get the
Carman-Kozeny relation for the permeability
k=
φ3
φ3
=
2S 2 τ 2
KS 2
(4.57)
where we have introduced the Carman relation given by the Kozeny constant
K0 times the turtosity τ squared.
Now in combination with a dose of experiments the Carman-Kozeny constant K is found to be approximately 5 for a random packing of spheres. With
the specific surface area found to be S = 3(1 − φ)/a for spheres of diameter
a packed at various porosities, the Carman-Kozeny relation for permeability
of porous media is,
φ3
a2
k=
.
(4.58)
9K (1 − φ)2
The porosity dependence in k is both experimentally [79] and numerically
[80] tested for Stokes equation of porosities less than 0.5.
18
The turtosity is defined by the ratio between the effective length and the sample length.
It can be deduced from measurements of the electric resistivity of samples saturated with
an electrolyte.
52
4.10. FLUIDIZING GRANULAR MEDIA
4.10
Fluidizing granular media
In this section I will describe how viscous flow through porous media will
take the porous packing from a solid state to a liquid state as described in
section 3.4. This process is known as fluidization.
Fluidization of granular media is a process similar to liquefaction whereby
a granular material is converted from a solid-like state to liquid like state. As
I described in the section of liquid granular media, the term fluid granular media bears some ambiguity19 . Within geological systems is the the principles
of gas flow induced fluidization well established (see e.g. review by Freundt
and Bursvik 1998 [22]). Fluidization occurs when the gas flow through a
bed of particles provides sufficient drag to support the buoyant weight of the
bed. Kunii and Levenspiel [25] identifies the minimal fluidization velocity
Umf of the imposed gas. Through a balance of inertial and viscous forces
and acceleration and bouyancy when the gas-solid mixture liquiefy,
1.75
s̃φ3f
dUmf ρg
µ
!2
150
dUmf ρg
+ 3 (1 − φf )
s̃φf
µ
!
=
d3 ρg (ρg − ρs )
g,
µ2
(4.59)
where s̃ is the spherity, φf is the porosity at fluidization20 , d is the mean
particle diameter, and ρg and ρs is the gas and solid density. The minimal
fluidization velocity is the fluidization velocity in a one dimensional setting.
At low gas velocities v < Umf , the gas will move upwards through the
bed via the empty spaces between the particles. The aerodynamic drag on
each particle is low so the bed will remain fixed, or solid. By increasing the
velocity the aerodynamic drag forces will counteract the gravitational forces.
Often the bed will expand in volume in this phase as the particles might
move away from each other. At a critical velocity the upward drag forces
will exactly equal the downward gravitational forces, causing the particles
to become suspended within the fluid. At this velocity the bed is said to be
fluidized and the granular media will exhibit fluid like behavior. A further
increase of gas velocity the particles no longer form a bed and hence go into
a gaseous phase.
A fluidized bed of solid particles behaves as a liquid, like water in a
bucket. The bed will conform the volume of the chamber, the surface will
be perpendicular to gravity and objects will float on its surface wobbling
19
The question was when we could define the granular media to be fluidized. The normal
approach is that the granular media is fluidized once it flows. Other define the bed to be
fluidized when it behaves as a liquid, e.g. does reduction of topography, buoyancy effects
are important etc. In this thesis I define the media to be fluidized once it flows.
20
The porosity just before fluidization can deviate from the porosity where no gas is
forced through the bed due to dilatancy.
53
CHAPTER 4. LIQUID FLOW IN POROUS MEDIA
up and down. When fluidized, the particles can be transported like fluids,
channeled through pipes. Due to the rapid dissipative effects, the granular
quickly relaxes into its solid phase again when the external energy injected
stops.
4.10.1
Classical fluidization criteria
In this section two hypothesis that might explain the onset of fluidization is
presented.
First hypothesis of fluidization
A common hypothesis [5] of fluidization that we worked along for several
months was the assumption that fluidization occurs when the pressure at
depth equals the lithostatic weight of the overlaying column. So by balancing
the forces between the lithostatic pressure and fluid pressure the transtion
between static and fluidized granular media will be found. Stated through
the force balance, we expect from this hypothesis that fluidization occurs in
this transition,
pf Ainl = Fg ,
(4.60)
where pf is the fluid pressure at the transition, Ainl is the area of the inlet
and Fg is the gravitational force acting on the body of the fluidized zone.
Second hypothesis of fluidization
The second hypothesis of when fluidization of the granular media occurs
is when the gravitational force equals the viscous drag force minus the the
boyancy FD = Fg − Fb [22]. Since the density of the air is much smaller
than the density of the grains we neglect the boyancy term in the following
argument. Now if the fluids flow past a bed of spheres, the viscous drag force
on each sphere is given by
Fvisc = CD FD ,
(4.61)
where CD is the drag coefficient21 . The drag coefficient is larger than 1 since
the neighboring particle would increase the drag onto the sphere of interest.
The drag coefficient for porous media are given in [32] to be a function of
2
a2
permeability and porosity, CD = 9(1−φ)
.
k
21
Examples of drag coefficients CD : Smooth brick - 2.1, a typical bicycle plus cyclist 0.9, Chevrolet Corvette mod. 2005 - 0.29, and SAAB-93 mod. 2003 - 0.28! In comparsion
with a porous media with a drag coefficient of ' 107 (off the formula).
54
4.10. FLUIDIZING GRANULAR MEDIA
The viscous drag on a sphere moving with velocity v through a infinite
viscous fluid is previously found (equation 4.29). Stokes drag force on a bed
of particles is the drag on a single spere times the drag coefficient,
Fvisc = CD 6πµav.
(4.62)
3
The gravitational force is Fg = mg = ρ 4πa
, where the last term is the volume
3c
occupied by a single particle. By balancing the gravity force and viscous drag
force neglecting the bouyancy force and solve for the flow velocity we get,
vfsb = kCK
ρg
' 1.8m/s
µ
(4.63)
Now this is the velocity necessary to lift the top layer of beads. The inlet
fluidization velocity through the inlet can now be found through mass conservation. This can be done if an expression between the inlet and surface
area can be found. Such an expression will be derived in section 7.3.
In the next part of the thesis an experimental study of viscous flow induced fluidization of granular media will be presented.
55
CHAPTER 4. LIQUID FLOW IN POROUS MEDIA
56
Part III
Experiment
57
Chapter 5
Venting in the laboratory
This chapter will start by a presentation of the experimental setup and the
material before I proceed by explaining how the experiment is performed.
Then I will give a presentation of the results.
In the presented set of experiment the conditions for flow localization and
fluidization of granular media is examined.
Hypothesis 1 is that air flow induced fluidization of granular media occurs
when the viscous drag of the air equals the gravitational force on grains. In
the fluidization process the flow focuses through a high permeable zone above
the inlet. The high permeable zone is from now on referred to as conduit.
There is also another way of thinking of fluidization. The second hypothesis is that fluidization occurs when the imposed fluid pressure at depth
equals the lithostatic pressure of the overlaying sediments.
When controlling the imposed flow velocity and the fill height of the bed
we are able to develop a phase diagram of when fluidization occurs. By
calculating the fluidization velocity from hypothesis 1 and 2 and comparing
it to the measurements one of the two hyptheses will be supported.
5.1
Experimental setup
The experiments were done on a vertically oriented Hele Shaw cell as seen in
figure 5.1. The cell is illuminated from behind by a light box to improve the
observations. The air inlet, with an inner diameter of 3.8 mm1 is placed 6.3
1
The diameter of the inlet is 1.6 mm smaller than the hose connecting the flow meter
to the inlet. In the calculations for the flow velocity of the compressible flow through the
hose the diameter of the hose is assumed to be constant. The narrowing of the hose at
the inlet increases the flow velocity at the inlet. The length of the inlet is about 10 cm,
which is about 1/10 of the total hose length. This effect is neglected in the calculations
for the flow velocity.
59
CHAPTER 5. VENTING IN THE LABORATORY
Figure 5.1: Figure A showing the experimental setup of the venting experiment. In figure B the Hele Shaw cell is shown as seen from the camera.
cm into the cell to prevent the air flow to focus along the walls. Compressed
air was used as the analogue material to induce the fluid pressure into the
bed consisting of dry2 Beijer glass beads with a diameter between 420 and
840 µm.
To measure the flow velocity and pressure difference we use Omega FMA1610 mass flow meter and Omega pressure sensors. The pressure sensors were
placed by the inlet and at the top of the bed. The flow measurements were
performed in the tube about 1 meter from the inlet. Labview(T M ) was used
to simultaneously log the pressure and flow measurements and the pictures
were captured by a high speed and resolution black and white Jai CV-M4
camera with a Nikon AF Nikkor 20 mm lens (10 frames per second).
The velocity and pressure measurements were processed and all the plots
are generated using Matlab. The flow measurements were transformed from
standard liters per minute at the flow meter in the hose to velocity at the
inlet. The pressure measurements were transformed from PSI to Pa.
5.1.1
The material
In this section the physical properties are given for the material we used in
the experiment. The air at a standard atmosphere has a density of ρair =
1.2 kg/m3 , a compressibility of Kb = 7.04 · 10−6 Pa−1 , and dynamic viscosity
2
The beads have low cohesion, in the order of ∼10 Pa. There has been a great debate
during the last year in which order of magnitude the cohesion is to be expected in the
experiments.
60
5.1. EXPERIMENTAL SETUP
µ = 17.6 · 10−6 Pas. The Youngs modulus of the spherical glass beads are
given to be E ' 71GPa [39].
A set of experiments were performed to find the density and porosity of
the glass beads. The measurements were performed on a cylindrical container
with liter marks on the side. The beads were filled through the exact same
funnel as in the other experiments. To induce the same stress field and
packing density for all experiments.
Figure 5.2: A closeup picture of the Beijer glass beads used in the experiment
with a diameter d=420-840 µm. Picture by light microscope, S. Hutton.
However, it was found that the porosity depended on the rate at which the
glass beads were filled. If the glass beads were poured in as quickly as possible
through the funnel, the porosity was measured to be φq = 0.400 ± 0.005. By
adding the beads slowly, the porosity was found to be somewhat less, φs =
0.370 ± 0.005. It takes some time for the single bead to settle properly.When
beads are quickly poured in, they “jam” before they settle properly.
In the following it is assumed that the porosity is independent of the geometry difference between the HS-cell and the cylinder on which the measurements were done and that bulk porosity lies in the region between the slowly
and quickly poured porosities. The average of the two is thus assumed to be
the porosity of the setup, φ = 0.385 ± 0.005.
The density of the glass is measured by precise measurements of the
weight of a certain volume of glass beads. The sample was filled in the
same way as described in the previous paragraph, both quickly and slowly.
By dividing the volume occupied by the beads with the weight and packing
(c = 1−φ) of the same beads, the density through ten experiments was found
to be ρ = 2460 ± 20kg/m3 . A closeup picture of the glass beads is shown in
figure 5.2.
The diameter d of the glass beads is given from the producer to be in
61
CHAPTER 5. VENTING IN THE LABORATORY
the range from 420 to 840 µm. The distribution of sizes within this range is
unknown. We assume the average grain size to be d = 630µm.
5.1.2
Air supply
The total pressure drop across the whole experimental setup is chosed from
the compressor to be ∆psetup ' 2 bar. This is much larger than the pressure
drop across the bed (∆p ' 1 PSI), so
∆p
1
' .
∆psetup
30
(5.1)
When this is the case, we do not expect the supplied flow velocity to depend
on what happens inside the bed.
This is supported by the fact that the flow velocity does not increase
even when a high permeable zone forms from the inlet to the surface (see
figure 5.3). We can thus say that the supplied air flow velocity does does not
depend on the physical processes happening inside the bed. This makes the
imposed air velocity a suitable candidate to be used in a phase diagram.
The pressure measurements are highly dependent of the processes occurring in the bed. As the high permeable zone (i.e. pipe) forms, the pressure
difference between the inlet and surface drops. Thus the pressure versus velocity measurements can be used to quantify how the bulk air resistivity of
the bed depends on flow velocity.
Finding the velocity at the inlet
The measurements were given to us in litres per minute (Q̃m ) in the hose
about L = 1.1 m from the inlet. Transforming to SI-units, the flow velocity
Q̃m
is given by Qm = 60·1000
, given in [m3 /s]. The theory and calculations in
this thesis is based on the velocity of the air at the inlet vinl = Q/Ah . Air is
compressible, thus the pressure difference through the hose has to be taken
into consideration when calculating the velocity at the inlet.
For a hose with elliptical cross section the equation for laminar flow was
solved analytically in Landau & Lifshitz Fluid mechanics p. 53 [81]. Considering the pressure drop through the hose ph with circular cross sectional
area (major and minor axis equals the radius), the equation given in Landau
& Lifshitz may be rewritten
ph =
8 Lµ
Q,
π rh4
62
(5.2)
5.2. PERFORMING THE EXPERIMENT
where the radius of the hose rh = 2.7 ± 0.2 · 10−3 m is assumed constant3 .
Due to compressibility, will the density of the air depend on the pressure
drop across the hose, as given in section 5.7,
(5.3)
ρm = ρinl exp(Kb ph ).
Assuming mass conservation from the measurments location to the inlet,
Ṁm = Ṁinl , the volume flow rate at the inlet is is given by,
ρm
Qinl =
Qm .
(5.4)
ρinl
An expression for the flow velocity may now be obtained,
vinl
!
8 Lµ
Qm
Qm .
= 2 exp Kb
πrh
π rh4
(5.5)
The numerical value of equation 5.5 transform the flux measured in the hose
Q̃m to the flow velocity at the inle. It is given by,
vinl = 0.73Q̃m exp 1.08 · 10−4 Q̃m .
(5.6)
This equation dominates the numerical values of the y-axis in the phase
diagram.
The noise in the velocity measurements are measured to be ' 1.7 · 10−3
m/s. At fluidization this is in the order of ∼ 10−4 of the measured fluidization
velocity.
Pressure drop
The pressure drop across the bed of glass beads are given in PSI, Pounds
per square inch. Transforming to standard SI-units, Pascal, was done by
multiplying with the transformation coefficient Ctp = 145.04 · 10−6 Pa/PSI.
The noise within the pressure measurements are measured to be ' 5.7 ·
−6
10 Pa. At fluidization this is in the order of ∼ 10−2 of the pressure at
fluidization.
5.2
Performing the experiment
The experiments were performed by a monotonic increase of flow velocity
by slowly opening a valve by the flow meter. A monotonic increase was important across the phase boundaries. When crossing a phase boundary the
3
Hereby assuming no elasticity in the hose and neglecting the pressure increase when
the inlet is narrower.
63
CHAPTER 5. VENTING IN THE LABORATORY
packing of the bed is sufficiently changed to disrupt the reproducibility of
the phase boundary. While controlling the flow velocity we measured the
pressure drop across the bed. When plotting the pressure drop versus the
imposed flow velocity we could measure the bulk responce of the bed. By
comparing the changes in the bulk responce/resistivity to the pictures taken,
we could correlate the observed phenomenae to the flow-pressure measurements. We performed a systematic series of experiments by varying the fill
height and registering the imposed flow velocities at the onset of bubbling
and fluidizatoin. The data is plotted together in a phase diagram of the
documented features, see figure 5.7.
5.3
Results
Three distinct physical phenomena; linear Darcy flow, static stable bubble
above the inlet and fluidization of granular meia, were observed in the system when performing the experiment as previously described. These three
phenomena are recognized in the p(v) measurements in figure 5.3. In the following sections a presentation of the three phases will be given in the order
of their occurence when the air flow velocity was increased from zero.
5.3.1
Linear regime
At flow velocities v between 0 and vb , see figure 5.3, the pressure drop ∆p
increases linearly with flow velocity. In this linear regime Darcy flow in
porous media applies,
µh
∆p = av =
v,
(5.7)
kef f
where a is a constant of proportionality between the pressure and velocity
measurements (figure 5.3). µ is the dynamic viscosity of the imposed air, h is
the fill height of the bed, and kef f is the effective permeability of the bed. kef f
is the permeability measured in the given geometrical setup. The measured
permeability will later be shown analytically to depend on the fill height. A
height dependency other than the explicit fill height can be deduced when
plotting the slope a = d(∆p)/dv against different fill heights.
In Darcy’s law, which is a 1D model, it is expected that doubling the fill
heigth the value of the slope a would double when the viscosity µ of the air
and the permeability k of the bed is kept constant. This is not observed in
the measurements plotted in figure 5.4. It is observed that a five fold increase
of fill height increases the slope by a factor 1.5. This suggests some height
64
5.3. RESULTS
e
b
Bu
n
bli
Fluidization
ne
Li
e
rr
im
eg
gr
e
m
gi
a
vb
vf
Figure 5.3: The slope in this ∆p(v)-plot reveals the bulk response of the
bed. The different physical phenomena changes the bulk behaviour and thus
abrupt changes in the p(v) measurements is observed. In the first regime we
observe a linear relation between the pressure drop and velocity. This regime
enables us to measure the permeability from the Darcy’s law for fluid flow
through porous media. The second regime, at flow velocities vb < v < vf we
observe a stable static bubble above the inlet. At vf the bed fluidized and a
pipe is formed from the inlet up to the surface.
dependency hidden in the permeability or viscosity that due to the geometry
will depend on height.
A hot candidate of where this height dependence is hidden, is in the
measurements of the effective permeability. An analytical solution of this
will be given in section 7.3 where the geometrical considerations are taken
into account. This may be used to understand the height dependence of a,
the solution is plotted in figure 7.3. This effect will be discussed more in
detail in the discussion chapter. When increasing the flow velocity up to
about 18 m/s a static bubble forms above the inlet.
5.3.2
Breakdown of linearity
The breakdown of linearity in figure 5.3 at the critical flow velocity vb , appears
simultaneously with the formation of a static bubble above the inlet. The
65
CHAPTER 5. VENTING IN THE LABORATORY
320
dp/dv
300
280
3
a, [Ns/m ]
260
240
220
200
180
160
140
0.05
0.1
0.15
Fill height, [m]
0.2
0.25
Figure 5.4: Plot of the slope between the pressure and velocity measurements
for different fill heights in the linear regime. The plot reveal large fluctuations
in the measurements though a positive trend between the fill height and
slope. A five fold increase of fill height increases the slope by a factor 1.5.
This suggests some height dependency hidden in the permeability or viscosity
in the geometrical setting.
precense of the static bubble effects the bed in two different ways, that would
both reduce the pressure measurements. The bubble decreases the height up
to the surface and increases the affective surface available to flow locally
above the inlet. These two effects happens simultaneously and effects the
pressure measurements causing a breakdown in linearity as may be seen in
figure 5.3.
For fill heights below 12 cm a static stable bubble is not observed. At low
fill heights the bubble instantly grows to the surface and fluidizes the bed.
For fill heights above 12 cm the bubble is found to form at flow velocities
vb = 18.4 ± 2.3 m/s for all fill heights. The observations of the bubbling
velocity are done based on image analysis and the breakdown of linearity in
the pressure velocity measurements. The onset of bubbling is plotted for all
experiments in the phase diagram in figure 5.7
When increasing the flow velocity above vb , the size of the static bubble
grows in discontinuous steps giving enlarged fluctuations and the decreasing
slope in the pressure-velocity measurements. This is shown in the plot of a
typical experiment in figure 5.3.
The size and form of the bubble are found to be velocity dependent, not
66
5.3. RESULTS
time dependent. I therefore report having found a static, stable bubble forming above the inlet. A search through the physics literature on bubbling in
granular media did not reveal any articles where this phenomenon is reported.
5.3.3
Fluidization
As the flow velocity increases up to a well defined velocity vf , the previously
static bubble rapidly grows to the surface and fluidize the bed. A picture
series of the fluidization process can be seen in figure 5.5.
The flow velocity necessary to fluidize the bed vf is marked off and plotted
within the phase diagram. From the phase diagram it is found that the
fluidization velocity increases linearly with fill height through the following
function obtained from least squared method,
vf (h) = (167.4 ± 6.7)s−1 h + 3.7m/s.
(5.8)
The standard deviation of the slope is found by taking the square root of the
variance defined through,
V ar(a) =
Pn
(vf i − (ahi + b))2
,
P
(n − 2) n hi − ˆ(h)
i=1
(5.9)
i=1
which gives σ1 = 6.7 s−1 .
The fluidized zone is recognized by a conduit in the center where the
beads are flowing upward, and a zone of grain flowing downward to the
center between the pipe and the static zone. A picture of the conduit and
flow field can be seen in figure 5.6.
The transition between the fluidized and static zone is mapped for the
set of experiments and plotted in figure 5.8. The size of the fluidized zone
scales with the initial fill height for experiments with fill height between 5
and 20 cm, though a slight narrowing of the zone is observed for larger fill
heights. Wich is supported by measurements of the angle α2 that decreases
with fill height(figure 5.9F).
By neglecting the narrowing effect we assume that the fluidized zone z(x)
(mapped in figure 5.8) scales with height, it enables us to find how the totally
fluidized mass m of beads relates to the fill height.
A derivation of the mass of the fluidized zone is given in section 7.1.2. The
assumption that the fluidized zone scales with height enables us to conclude
that no non-linear effects such as Jansen wall effect plays any important role
in the experiments.
The linear dependence in the fluidization velocity versus fill height is
discussed in section 7.2.
67
CHAPTER 5. VENTING IN THE LABORATORY
Figure 5.5: Picture series of the transition from the static bubble to the
fluidized phase, ten frames per second. We see that the total time for the
fluidization process is 0.7 seconds. During the fluidization process the imposed velocity was kept constant. When the imposed flow velocity reaches
vf the viscous drag force on the bed of beads equals the weight of the overlying bed. The bubble can rapidly grow to the surface and mark the onset of
fluidization. Inward dipping of the surrounding strata can be seen since the
grains have a downward flow along the sides of the fluidized zone.
68
5.4. GEOMETRICAL MEASUREMENTS
Saddle
Totally affected
widht
α os
Crater zone
α3
α2
Static fluid
transition
Pipe zone
Downw ard flow
Conduit
Static zone
Figure 5.6: Picture of the fluidized phase of the experiment. The fluidized
zone consists of a conduit from the inlet to the surface and grains flowing
downward along the side. The flow field of the particles are sketched directly
onto the glass plate. The wall between the fluidized and static zone have low
pipe zone and a upper craterzone close to the surface.
5.4
Geometrical measurements
Several measurements were performed on the geometry of the fluidized zone.
These measurements might in give some interesting insight into the formation
of related structures formed by fluidization in geology such as Kimberlites
and htvc.
In figure 5.9A the width of the conduit is measured at half the fill height
from the inlet. The width of the conduit increases linearly with the fill height
h, as expected since the flow velocity necessary to fluidize the bed increased
linearly with height. The higher flow velocity the wider conduit is needed
to accommodate the flow. In this sense there is a competition between the
cross sectional area of the flow through the pipe and the flow velocity. The
Reynolds number above the pipe is quite large, thus it might be energetically
69
CHAPTER 5. VENTING IN THE LABORATORY
60
50
Measured v
f
Measured vb
Best fit of vf
Average of v
b
σ of v
1
f
σ1 of vb
Fluidizing
Fluid velocity, [m/s]
40
30
Bubbling
20
10
0
0.05
Static granular media
0.1
0.15
Fill height, [m]
0.2
0.25
0.3
Figure 5.7: Phase diagram showing the three different phases in the experiment. The measurements of the critical velocities vb and vf marks the
onset of bubbling and fluidization for the experimental series of fill heights.
The flow velocity necessary for fluidizing the bed increases linearly with the
fill height with the functional form vf (h) = (167.4 ± 6.7)s−1 h + (3.7)m/s.
The velocity necessary to form the bubble above the inlet is measured to be
vb = 18.4 ± 2.3 m/s.
70
5.4. GEOMETRICAL MEASUREMENTS
Figure 5.8: Mapped transition between the static and fluidized phase for
several experiments ranging from 5 to 20 cm. In A the transition between
the static and fluidized zone z(x) is plotted for several fill heights. In B
the transition z(x) is scaled with the initial fill height. The transition scales
almost linear with the fill height, except at fill heights below 12 cm. Two
measurements are shown here in this regime, 8 and 10 cm (red and green
line).
easier to increase the conduit width in stead of increasing the flow velocity
further.
In figure 5.9B the width of the conduit is plotted against the saddle width.
A linear relation between the two is observed. This observation might give
some insight into geologically related structures. The width of the conduit is
found to increase by a factor ∼ 0.06 times the saddle width. By measuring
the distance between the saddles in geology one can obtain estimates of the
width of the conduit.
In figure 5.9C the saddle width, defined in figure 5.6, is measured and
plotted against the fill height. The saddle is produced when the particles are
thrown upward to both sides through the vertical conduit above the inlet. A
linear dependence between the fill height and saddle width is reported, this
can also be seen from the mapping of the fluidized zone in figure 5.8. This
is reasonable when considering the phase diagram where it is found that the
velocity necessary to fluidize the bed increases linearly with height.
The angle between the horizontal and the angle at which the grains settles
along both sides of the saddle (inner and outer) was measured for the experiments. In figure 5.9D the angle is plotted against fill height. The angle did
not seem to relate to the fill height, thus also the flow velocity. The inner
saddle angle were measured to be αis = 22 ± 4o , while the outer saddle angle
71
CHAPTER 5. VENTING IN THE LABORATORY
αos = 21 ± 2o . These angles are interpreted as being the dynamical angle of
repose of the system, since it is the inclination at which the grains settle as
they fall onto the saddle (φr ' 22o ).
The total affected width of the fluidized zone was measured and plotted
against fill height in figure 5.9E. When the geological system is formed by
fluidization of the matrix, measurements of the total affected width could
potentially estimate the depth of the fluidized zone. It is found that the
totally affected width increases with a factor ∼ 2.1 times the fill height.
Measurements on the crater angle showed no height dependency with an
average of α3 '45o .
wt(h)= 0.086842*h + 0.24474
2
1.5
1
0.5
C
2.5
Wc(ws)= 0.062775*ws
40
W(h)= 1.1804*h + 5.4545
35
2
Saddle width (Ws), [cm]
B
2.5
Conduit width (Wc), [cm]
Conduit width (wc), [cm]
A
1.5
1
0.5
30
25
20
15
10
0
D
0
0
5
10
15
Fill height (h), [cm]
20
25
E
40
0
25
20
15
10
25
30
35
Saddle width (Ws), [cm]
10
12
14
16
18
20
Fill height (h), [cm]
22
24
26
0
5
10
15
Fill height (h), [cm]
20
25
35
30
40
30
25
20
20
0
W(h)= −0.5078*h + 32.7271
15
10
8
5
40
α2, [deg]
Total width (wT), [cm]
Angles (α), [deg]
os
20
F
50
30
15
wt(h)= 2.1069*h + 4.4597
is
α
10
60
α
35
5
0
5
10
15
Fill heigth (h), [cm]
20
25
10
0
5
10
15
20
Fill height (h), [cm]
25
30
Figure 5.9: In figure (A) is the width of conduit plotted against the fill
height. In figure (B) the width of the conduit is plotted against the width of
the saddle. The distance between the saddles plotted against the fill height
in figure (C). Linear relations is observed in figure (A-C). In figure (D) are
the inner - and outer saddle angle plotted for different fill heights. These
are angles are measured from the horizontal direction and does not seem to
depend on h. In figure (E) the total width of the feature is plotted against
the fill height. In figure (F) the angle between the fluidized and static zone
at half fill height is plotted against the initial fill height. By increasing the
fill height the fluidized zone steepens.
72
5.5. DIMENSIONAL ANALYSIS
5.5
Dimensional analysis
For historical reasons were the experiments inspired by observations on htvc.
In this section I present the application of the model to natural systems.
Dimensional analysis enables us to relate the physical properties in dimensionless ratios. With the functional form of the transition for fluidization
known (from the phase diagram) the conditions for venting can be plotted
for realistic values for depth (i.e. fill height h), permeability k, and flow
velocity v.
5.5.1
Dimensional analysis
In this section we develop scaling relations that maps the results from the
laboratory onto geological scales. Non-dimensional ratios in the phase diagram for the critical air velocities can be found through dimensional analysis.
By deriving the diffusion equation of viscous flow through porous media hints
a reasonable group of parameters. Considering air mass conservation in a
static media, where the change of mass in a unit volume in unit time equals
the difference between the flux in and out of the same unit volume. Assuming
constant porosity, we can formulate the statement above through the density,
∂ρ
∂(ρv)
=−
,
(5.10)
∂t
∂z
where v is the flux in [m/s], ρ is the density in [kg/m3 ]. Darcy’s law stating
that the flux is proportional with the pressure difference,
k ∂p
v=−
,
(5.11)
µ ∂z
where µ is the dynamic viscosity of the fluid given in [Pas], k is the permeability in [m2 ] and p is the fluid pressure in [Pa]. Since we use air as the fluid
driving the formation, we have to consider the compressibility. Through a
first order Taylor expansion we can relate the change in density to a change
in pressure through the compressibility via
ρKb dp = dρ,
(5.12)
where Kb is the compressibility, given in [Pa−1 ]. Combining the compressibility, Darcy’s law and mass conservation gives us the normal diffusion equation
of the form
!
k ∂p
dp
∂
(5.13)
=
dt
∂z Kb µ ∂z
!
∂ k ∂p
=
p
.
(5.14)
∂z µ ∂z
73
CHAPTER 5. VENTING IN THE LABORATORY
A reasonable group of parameters to be used in the dimensional analysis could
then be k/µKb , given in [m2 /s]. Other parameters that might be important
for the critical fluidization velocity could be the lithostatic pressure at the
inlet ∆ρgh, given in [Pa], the inter particle cohesion C, in [Pa], the Youngs
modulus of the beads E, in [Pa], angle of friction θif , and height h, in [m]. The
critical velocity of fluidization vf is an unknown function f of the parameters
vf = f
!
k
, ∆ρgh, C, E, θif , h .
µKb
(5.15)
Since E Pf the resulting elastic strains would be inconsequential for the
observed dynamic. By small variations of grain size and cohesion we know
from [referanserreferanser] that the angle of friction α is preserved in granular
media. We can therefore skip these terms and chose k/µKb , ∆ρgh and h as
our fundamental units. The fluid velocity vf is given with a unknown function
of f with the following set of parameters,
k
vf =
f
hµKb
∆ρgh
C
!
=
k
f (α1 ).
hµKb
(5.16)
The functional form of f is to be found through the set of experiments.
b
In figure 5.10 we show a non dimensional plot of hµK
vf as a function of
k
∆ρgh
. With the functional form of vf (h) given from the laboratory we find
C
that f (α1 ) is a parabolic eqaution with the exponent 2 (see figure 5.10). This
enable us to relate the critical velocity for fluidization to the height, density difference, cohesion, permeability, compressibility and the gravitational
constant to the natural environments.
The values following values are used in the plotted phase diagram: cohesion C = 10 Pa, density difference ∆ρ = 2000 kg/m3 , permeability k =
1.33 · 10−9 m2 , compressibility Kb = 7 · 10−6 Pa−1 and gravity g = 9.81 m/s2
and then the measurements of the height and flow velocity. These values are
of course constantly up to discussion.
By using these values we find that
∆ρgh
f (α1 ) = 16.04α12 = 16.04
C
!2
.
(5.17)
By using the functional form that we get of f (α1 ) we can vary some of the
variables e.g. the permeability and height.
74
5.5. DIMENSIONAL ANALYSIS
Non dimensional phase diagram
2
Fluidization velocity, v
f
Bubbling velocity, vb
Functional form of f
1.8
1.6
σ for f
α
α
1
1.4
Fluidized media
1
f
b
v hµK /k, [1]
1.2
0.8
0.6
Bubbling
0.4
0.2
0
Static media
0
100
200
300
400
∆ρ gh/C, [1]
500
600
700
Figure 5.10: Dimensionless phase diagram of the documented behavior. The
transition marking the onset of fluidization is marked by the solid line. Along
the y-axis is f (α1 ) plotted against ∆ρgh
, as described in the text.
C
75
CHAPTER 5. VENTING IN THE LABORATORY
5.6
Venting in natural systems
Armed with the dimensionless variables and the functional form of f (α1 )
measured in the lab, we now can plot how all different variables relate to
each other to determine the conditions for venting in all natural systems. As
discussed earlier in section 5.3.2 we concluded that it is the flow velocity that
is important for fluidization in this case and not the pressure. As we remember, this was firstly due to the feedback of the system in the bubbling phase
leading to the non-linear behavior in the p(v) measurements. Secondly; that
theoverlaying sediments liquified when the viscous drag equal the gravity.
When varying the depth of the source (i.e. fill height) we find that it is
mainly the permeability that varies the most of the parameters in equation
5.16. By keeping the gravity g = 9.81 m/s2 and density difference ∆ρ = 2.5 ·
103 kg/m3 constant we can vary the height along the x-axis and permeability
along the y-axis. By varying the fluid that fluidizes the media between air,
water and magma along the horizontal axis and three different host rocks
along the vertical axis we develops a four dimensional phase plot. The group
of physical parameters that determines the physical property of the viscous
fluid are the product ι of viscosity µ and the compressibility Kb . The group
is summarized in the following litst:
• For air we use µair = 1.7 · 10−5 Pas and Kbair = 7.0 · 10−6 Pa−1 , giving
χair = 1.3 · 10−10 s−1 .
• For water we use µwat = 8.9 · 10−2 Pas and Kbwat = 4.55 · 10−10 Pa−1 ,
giving χwat = 4.0 · 10−11 s−1 .
• For magma we use µmag = 1 · 103 Pas and Kbmag = 1 · 10−11 Pa−1 , giving
χmag = 1 · 10−8 s−1 .
The values for the magma changes over several orders of magnitude dependendt of the rate of crystallization etc. The cohesion of the host rock is set
to vary over three orders of magnitude: C1 = 0.1 MPa, C2 = 1 MPa and
C3 = 10 MPa.
With these limits on the physical parameters we can now vary the depth
h from laboratory scale to the size of kimberlites, from 1 m to 105 m. The
permeability is set to vary ten orders of magnitude from k = 10−20 m2 to
k = 10−10 m2 . Now by using f (α) we can calculate the critical fluidization
velocity that breaks the pore space and form piercement structures with this
set of independent variables C, χ, µ and h. This is done in figure 5.12.
Boiling and rapid maturation of organic materials occurs in aureoles
within volcanic basins (as described in the introduction) due to the heat
76
5.6. VENTING IN NATURAL SYSTEMS
0
−500
Fluidization
Depth, [m]
−1000
−1500
Static
−2000
v = 1 m/s
v = 10 m/s
v = 100 m/s
−2500
−3000
−15
−14.5
−14
−13.5
−13
−12.5
Logarithm of permeability, [m2]
−12
−11.5
−11
Figure 5.11: With the functional form of the fluidization function f (α1 )
given from the experiment and dimensional analysis, the transition for flow
localization is plotted when varying the depth between 0.1 to 3 km. Three
flow velocities are applied, vf 1 =1, 10, and 100 m/s. The transition shifts
towards right for higher velocities, hence fluidization can occur at higher
permeabilities at a given depth. Flow localization and fluidization within
the localized zone occurs at low permability and shallow depths. The ’dash
dotted’ lines marks one standard deviation within the experimental model.
from magmatic sill intrusions. The processes of boiling and maturation increases the volume (and pressure) of the gas within the overlaying sediments.
From arguments in chemistry, the rate at which this volume increases could
potentially be found. With the volume increase given, constrains of the flow
velocity v through an imagined horizontal surface above the aureole can be
found.
With the flow velocity given the conditions for venting in natural systems can be found from f (α1 ) when varying the depth h and calculation
the permeability k and keeping the cohesion, density difference and gravity,
viscosity and kompressibility constant. In figure 5.11 the transition between
static granular media and fluidization is plotted when the flow velocity vf =
[1, 10, 100] m/s.
77
2
Oil host rock10
Granite
20
10
Permeability, [m ]
Gravel
0
10
8
1
10
2
3
4
10 10 10
Depth, [m]
10
5
10
Host rock 1, Water
10
15
0
10
6 4 2 0 2 4
Velocity, log10(v)
1
10
4
20
10
0
10
1
10
2
3
4
10 10 10
Depth, [m]
10 8 6 4 2 0 2
Velocity, log10(v)
5
10
10
15
10
20
Host rock 2, Water
10
1
10
2
3
4
10 10 10
Depth, [m]
10 8 6 4 2 0
Velocity, log10(vf)
10
10
15
0
10
1
4
10
4 2 0 2 4
Velocity, log10(v)
6
10
2
3
10 10 10
Depth, [m]
5
10
20
5
10
2
1
10
2
3
4
10 10 10
Depth, [m]
6 4 2 0 2
Velocity, log10(v)
5
10
4
Host rock 2, Magma
10
10
10
15
10
20
0
10
1
10
2
3
4
10 10 10
Depth, [m]
12 10 8 6 4 2 0
Velocity, log10(v)
5
10
2
f
Host rock 3, Water
Permeability, [m ]
Host rock 3, Air
0
0
10
10
f
20
10
20
10
10
10
15
10
20
Host rock 3, Magm
a
2
15
10
f
2
10
15
2
10
6
Permeability, [m ]
10
10
Host rock 1, Magm
a
10 8
Permeability, [m ]
10
4
2
10
10
2 0 2 4 6
Velocity, log10(v)
f
Permeability, [m ]
5
10
2
15
Host rock 2, Air
Permeability, [m ]
10
4
3
10 10 10
Depth, [m]
10
f
2
Permeability, [m ]
10
2
20
f
10
Diamonds
15
10
Permeability, [m2]
10
Host rock 1, Air
OIL
10
Laboratory
10
2
Permeability, [m ]
CHAPTER 5. VENTING IN THE LABORATORY
0
10
1
4
10
6 4 2 0 2
Velocity, log10(vf)
4
10
8
2
3
10 10 10
Depth, [m]
5
0
10
1
10
2
3
4
10 10 10
Depth, [m]
12 10 8 6 4 2
Velocity, log10(vf)
5
10
0
Figure 5.12: By varying the properties of the fluid that induces the fluidization and host rock properties (horizontally and veritcally respectively) I
present this set of calculated fluidization velocities. All the plots look remarkably similar due to the fact that we use the same functinal form of
f (α1 ) for all the plots, it is just a variation of the pre factor. Along the
x-axis of each plot the depth is varied from laboratory scale to kimberlites
and the permeability from granite to78gravel. The value of the fluidization
velocity has to be read off the colorbar for each plot.
5.7. FLOW OF COMPRESSIBLE FLUIDS IN GRANULAR MEDIA
5.7
Flow of compressible fluids in granular media
For compressible fluids (air) the density of the fluid is related to the pressure
through an equation of state of the form, as e.g. given in [82]
ρ = ρ0 exp (Kb (p − p0 )) ,
(5.18)
where Kb is the bulk modulus. The bulk modulus of a compressible media is
defined by assuming that the density is linearly dependent of the pressure,
so
1 ∂ρ
.
(5.19)
Kb ≡
ρ ∂p
This definition of the compressibility is ambiguous in the way that it does not
take into account whether the compression is adiabatic4 or isothermal5 . The
compressibility of air is given to be Kb ' 7 · 10−6 P a−1 . Taking the gradient
of the equation of state 5.18 and solve for the pressure gradient yields,
∇p =
∇ρ
Kb ρ
(5.20)
which by used in equation 4.42, giving us a differential equation for the
density
∂ρ
k
=∇
∇ρ = ∇D∇ρ.
(5.21)
∂t
φµKb
D has dimensions [m2 /s] and is interpreted as a “diffusion constant” for the
density. This equation is a linear differential equation equivalent to diffusion
equations and heat conduction equations.
A numerical solution for stable flow through the porous media, by using
Laplace equation, is found by using Finite Element Method. The boundary
conditions for the experiment can be seen in figure 5.13.
The isocontours of the pressure field is plotted in figure 5.14. It can be
seen from the plot that the the pressure in this setting has a non linear
dependence. The flow velocity is given by the gradient of the pressure field
giving a flow perpendicular to the iso contours shown in the plot in figure
4
An adiabatic process happens so fast that no heat flows out of (or into) the gas. For
an ideal gas, the pressure times the volume to an exponent γ is constant through the
process, P V γ = constant.
5
In an isothermal process the temperature is kept constant within the gas through the
compression. For isothermal compression the heat leaves the gas. For an ideal gas the
pressure is given by P = N kVb T , which gives a concave-up hyperbola.
79
CHAPTER 5. VENTING IN THE LABORATORY
Figure 5.13: Boundary conditions used in the numerical modelling. It deviates from the experiment since the inlet is placed about 6 cm into the cell
to prevent the air to focus along the walls and h is measured from the tip of
the inlet to the surface.
5.14. It can be seen that the flow velocity along the surface is larger in the
center above the inlet than on the sides.
The bed will fluidize when the fluid velocity at the surface reaches the
necessary velocity to lift the top layer. This would occur in the center above
the inlet.
A proper analysis of the fluid velocity at the inlet given the fluidization
velocity on the top of the bed with the correct boundary conditions is is not
yet performed. This could be the way of reproducing the measured values in
the phase diagram in figure 5.7.
80
5.7. FLOW OF COMPRESSIBLE FLUIDS IN GRANULAR MEDIA
Time: 300 years.
1.8
1.6
1.4
1.2
2
1
1
1.5
1
0.5
y
0.5
0
0
x
Figure 5.14: This is a plot of the flow velocity field through the porous media
found by Finite Element Method. In this figure I have solved the diffusion
equation for the pressure and waited long enough so the time derivative of
the velocity is zero. It is in essence the same as solving the Laplace equation
for the pressure, ∇2 v = 0 assuming the permeability is constant through the
process. The isocontours are curved, and the larger flow velocity along the
surface is found in the center.
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CHAPTER 5. VENTING IN THE LABORATORY
82
Chapter 6
Additional experiments
In addition of the presented set experiments, additional experiments have
been performed. In these experiments we have not performed any systematic
variation of any physical quantities, thus no new fundamental knowledge can
be drawn out of the experiments. They serve as examples of some of the vast
peculiar properties that granular media exhibits.
In this chapter we will qualitatively describe and discuss the processes
occurring in various experiments. These experiments serve as examples of
natural processes occurring in nature. We therefore could potentially call
this chapter a Geological field study of the experiments.
Due to the large heterogeneities in nature some of the presented experiments approach a geological setting better than the setup described in the
previous chapter. In the presented examples processes such as fracturing,
fluidization, and deformation occur which are all prominent in nature. The
experiments gave us an enlarged understanding of the naturally related processes.
The set of experiments and discussion is done in corporation with Berit
Mattson. The experiments are performed on the same setting and in the
same way as described in section 5.2.
6.1
Transition from fluidization to fracturing
A fundamental observation is that cohesive material tend to fracture in stead
of fluidize. A more quantitatively statement could be the following; In materials where on grain level, the cohesive forces are more important than
gravitational forces fracturing will occur. This effect can be achieved by reducing the grain size (use of clay) or by adding small amounts of fluids to the
glass beads. The fluid forms liquid bridges between the grains that through
83
CHAPTER 6. ADDITIONAL EXPERIMENTS
its surface tension induce the cohesive force. In the following three sections I
will first present an example of fluidization followed by two examples where
fracturing occurs.
Figure 6.1: Experiments performed on a bed of dry glass beads. The series of
pictures show the evolution from Darcy flow at low flow velocities (a), through
early bubble phase above the inlet (b), formation of a V-structure and doming
on the surface (c), onto fluidization of the bed (d-f). No fracturing, only
fluidization is observed in these experiments.
84
6.1. TRANSITION FROM FLUIDIZATION TO FRACTURING
6.1.1
Experiments on dry glass beads
These experiments are the same as are discussed in the previous chapter. A
series of experiments were performed by varying the fill height. At low flow
velocities, no visible movements was seen in the matrix indicating Darcy flow
(figure 6.1a).
In the pressure versus velocity curve was this regime recognized by a
linear relation. No feedback on the pores space. At higher imposed fluid
velocities the pore space could no longer accommodate the imposed flow. A
feedback on the matrix was observed as a static stable bubble forms above
the inlet (figure 6.1b). The bubble size was found to grow with the imposed
air velocity. At a larger bubble size (figure 6.1c) a V-structure formed from
the air inlet up to the surface. This also caused a dome structure at the
surface. At a certain flow velocity, that was found to increase linearly with
fill height (figure 5.7), the bubble rapidly grew to the surface and fluidized
a zone of the overlaying sediments (figure 6.1d-f). An eruption occurred on
the surface fed by a conduit from the inlet. The V-structure that was formed
due to the static bubble was found to be conserved through the process of
fluidization.
6.1.2
Experiments on a bed of clay
A couple of experiments were performed with a homogeneous packing of clay
(figure 6.2a). In these experiments, no Darcy regime were found to exist
at all because fracturing of the matrix occurred immediately after air flow
was injected into the sediment (figure 6.2b-e). The reason is that in low
permeable rocks, any induced fluid will rapidly increase the pressure. As the
air flow localized, the fracture stabilized into one position.
By increasing the flow velocity, another pulse of fracturing occurred with
surface eruptions (figure 6.2f). An increase of flow velocity seemed to pull the
surface eruption towards the centre above the inlet in rapid pulses. Inside
the stabilized fracture (figure 6.2h), fluidization and flocculation of lumpy
clay particles was observed. A closer picture of this process can be seen in
figure 6.2i.
An observation done on the fracture network is that a fracture might
collapse and close after the pressure is released through second fracture.
85
CHAPTER 6. ADDITIONAL EXPERIMENTS
Figure 6.2: Fracturing process in a bed of low permeable clay. the fill height
of this sample is 12 cm. The fractures developed in image a-e developed
as the experiment started. These frames are separated by 1/10 second. A
further increase of flow velocity developed the fracture network onto forming
a stable conduit from the inlet to the surface. Flocculation and fluidization
of clay particles are observed within the conduit in h, which can be studied
in detail in figure i.
6.1.3
Experiments on wet bed of glass beads
A second way of inducing cohesive forces was to drain water through the
porous bed of glass beads. By filling the Hele Shaw-cell by dry glass beads
using the same method as before, the porosity should be the same as in the
dry experiments. Inhomogeneities due to capillary rise of water within the
bed were prevented by pressuring air from the top of the bed through the
bottom of the cell. The excess water that did not wet between the beads
were drained off giving an almost “homogeneous” cohesive bed. It has been
showed in [37] that the water forms small liquid bridges between the grains
causing the cohesive forces.
The cohesion could not be altered in any controllable way, nor was its
value known by using this technique. The reason for doing it was that it was
relatively simple and straight forward way of inducing the cohesivity.
When increasing the flow velocity into the bed, we saw a striking difference in the behaviour compared to the dry case. No bubbling was observed,
86
6.1. TRANSITION FROM FLUIDIZATION TO FRACTURING
but in stead we saw that formation of mode 1 fractures. Though when letting
the air flow through the material for some time (∼100 s) drying effects and
the formation of a bubble was observed.
Figure 6.3: Fractures in a bed of glass beads containing small amounts of water that induced the cohesive forces. The presented experiment consisted of a
homogeneous packing where the dark layers were spray painted glass beads.
Due to the relatively high permeability, a non-deformation regime at low flow
velocities was observed (a). The initial fracture started out as a horizontal
fracture (b) suggesting that the largest direction of stress was horizontal.
The fractures propagated with an inner angle of 90o (d-f). At higher flow
velocities preferential cracks developed within the initially formed structure,
finally evolving to the final conduit (i). Within the final conduit, fluidization
and flocculation of different sized lumps of glass beads were observed.
From figure 6.4 the velocity necessary to form the first fracture vf rac and
break the cohesive bonds above the inlet were found to increase with height.
The velocity was found to be vf rac = [5.6, 6.8, 8.0] m/s, for the fill heights
h = [0.10, 0.14, 0.18] m respectively. This suggests that the average of
the maximal and minimal stress direction increased with fill height1 . When
comparing to the bubbling velocity (vb = 18.5±2.0 m/s) we see that it is
actually “easier” to form the first fracture than the bubble. When the final
1
This might support the fact that the Janssen wall effect is of negliable importance.
87
CHAPTER 6. ADDITIONAL EXPERIMENTS
conduit forms is not well defined for these experiments. We therefore chosed
to defined the final conduit when the pressure difference across the bed had
dropped to one tenth of the maximal. A plot of the pressure velocity measurements of a fracturing experiment can be seen in figure 6.4. Any height
dependency in the necessary fluid pressure to form a fracture is not observed.
Due to time constraints and technical problems only three proper experiments were performed. Thus the deductions that is made from these
experiments are related to large uncertanties.
Fracturing vs. fluidization
50
Fluidization velocity, dry glassbeads
Bubbling velocity, dry glassbeads
v at first fracture
v at final conduit
45
4000
Fluidizing
3500
35
Pressure (p), [Pa]
Flow velocity (v), [m/s]
40
4500
30
25
Bubbling
20
15
10
Static
0
0.05
0.1
0.15
Fill height (h), [m]
First fracture
2500
2000
1500
Final fracture
1000
First fracture
5
3000
0.2
500
0
0.25
0
5
10
15
20
25
Flow velocity (v), [m/s]
30
35
40
Figure 6.4: Figure A shows a plot of the flow velocity necessary to form the
first fracture vf rac and final conduit against the fill heights for three different
experiments. The bed consisted of wet spherical glass beads. Measurements
p(v)-measurements of the fracturing process is showed for one specific experiment. The first fracture developes at about 8 m/s and the “final fracture”
or conduit forms at vfrac =35 m/s. Between these two imposed flow velocities the fracture network devolopes. A rapid drop in pressure measurements
marks the development of a new fracture.
6.2
Heterogeneous beds
In this section we observe deformation, fracturing and fluidization in heterogenous beds. The three first experiments presented in this section were
performed on a bed of glass containing a 0.3 cm layer of low permeable clay.
The fourth experiment in this section was performed on a bed of clay containing a 1 cm layer of glass beads and the fifth on large and small glass beads.
Several other experiments on glass beads with thicker clay layers (>0.3 cm).
These experiments flawed in the sense that the air focused along the walls,
thus no useful results were obtained. Different physical processes occurred
88
6.2. HETEROGENEOUS BEDS
dependent of on the position of the low permeable clay layer.
6.2.1
Experiment with a deep low permeable layer
At low flow velocities small vertical fractures were found to form within
the clay layer. The air flow localized through the fractures and pulled clay
particles into the overlaying layer of glass beads. This is seen as a decrease
in brightness. No bending of the clay layer was observed, due to the weight
of the overlaying bed.
Figure 6.5: Sequence of images from the experiment performed on a low level
clay layer within the glass beads. At low flow velocities, vertical fractures
developed in the clay layer. A static bubble formed above the inlet (b).
The size of the static bubble increases with flow velocity (b-c). At a given
flow velocity, the bubble expands and rapidly grew to the surface (d-g), thus
fluidizing the bed (h). The segregation phenomena is marked off in image h.
89
CHAPTER 6. ADDITIONAL EXPERIMENTS
An increase of the flow velocity formed a bubble above the inlet (figure
6.5b). The bubble was found to increase with flow velocity, and formation of a
V-structure above the inlet and a dome structure of the surface were observed
(figure 6.5c). A change in porosity in the glass beads were clearly observed
due to the brightening of within the V-shaped. At a critical flow velocity the
bubble rapidly grew to the surface causing fluidization (figure 6.5d-g). Due
to the relatively high cohesive clay layer, it was found to resist the erosion of
the fluidized particles in the conduit zone (figure 6.5h). Segregation of the
fluidized clay and glass bead mixture were seen as the clay particles tended
to settle along the edges of the V-shaped structure.
6.2.2
Experiments with a shallow low permeable layer
The experiment were done on a bed of glass beads containing a shallow layer
of low permeable clay. At low velocities the clay layer and the overburden
rise above the lower glass beads making a horizontal fracture (figure 6.6b).
This occurs due to the low permeability of the clay layer that increases the
air pressure to overcome the weight of the overburden. This effect was mainly
observed in the center region thus suggesting that small amounts of air penetrated along the edges of the cell. At higher flow velocities the horizontal
fracture grew, and deformation and bending of the clay layer was observed
(figure 6.6c-d). As described in the previous section, vertical fractures in the
clay layer and thus localization through these fractures brought clay fragments into the glass beads above.
Deformation and doming of the clay layer and the overburden glass beads
increased with flow velocity. Due to stretching, the edges of the dome yielded
before it collapsed (figure 6.6e-f) causing an eruption from the base of the
clay layer to the surface. The eruption propagated horizontally, mowing away
from the initial erupted dome through the center to wards the other side.
Parts of the horizontal original fracture was still intact when the propagation
ended (figure 6.6g). The eruption continued at both sides of the cell. Boiling
movements in the central parts between the end eruptions occurred. The
upper layer consisted of a mixture of clay and glass beads.
At higher flow velocities a static stable bubble formed above the inlet
while the two surface eruptions continued (figure 6.6h). The bubble and the
characteristic V-structure due to porosity reduction, could be seen. When
increasing the flow velocity further, the static bubble rapidly grew to the
surface and erupted up to the surface (figure 6.6i).
90
6.2. HETEROGENEOUS BEDS
Figure 6.6: Sequence of images from the experiments performed with a clay
layer at a high level within the bed of glass beads. At low velocities, a
horizontal fracture formed within the clay layer simultaneously with lifting
of the clay layer that made a dome structure (b-d). The dome bursted along
the edges and fluidized the top layer (e-g). An increase of flow velocity caused
the formation of a static bubble above the inlet (h), which grew to the surface
and fluidized the bed (i).
At higher flow velocities a static stable bubble formed above the inlet
while the two surface eruptions continued (figure 6.6h). The bubble and the
characteristic V-structure due to porosity reduction, could be seen. When
increasing the flow velocity further, the static bubble rapidly grew to the
surface and erupted up to the surface (figure 6.6i).
The mixture of particles segregated and the small clay particles concentrated along the edges of the V-structure / conduit zone. During the eruption
a distinct coloring from the clay on the glass beads within the conduit zone
were observed.
6.2.3
Experiment on two clay layers
This experiment were done on a bed of glass beads containing two layers
of clay (figure 6.7a). At low flow velocities a horizontal fracture occurred
between the upper clay layer and the bed below (figure 6.7b). No deformation
such as bending were observed, but small vertical micro fractures were found
91
CHAPTER 6. ADDITIONAL EXPERIMENTS
where the flow had localized and pulled clay particles into the overlaying
bed of glass beads. The horizontal fracture underneath the upper clay layer
expanded as the air flow increased. This formed a dome (figure 6.7c). The
dome structure collapsed, and an eruption propagated horizontally along the
surface away from the initial eruption (figure 6.7d-e). At higher flow velocities
a static bubble appeared just above the air inlet (figure 6.7e). Simultaneously
localized air flow brought clay particles into the overlaying bed of glass beads
through vertical micro fractures within the lower clay layer.
Figure 6.7: Sequence of images from the experiment performed with two
clay layers (a). At low flow velocities the upper clay layer lifted and formed
a horizontal fracture between the glass beads and the clay (b-c). The dome
bursted (d) and fluidized the top layer (e-f). The skewed V-structure (f)
is interpreted to form due to the air pocket in the erupted top layer (e).
Similar skewed structures are also documented by Mourgues et. al. in 2003
[83]. The static bubble rapidly grew (f-k) and led to the formation of a
conduit (l). Segregation is observed along the boundaries of the V-structure.
92
6.2. HETEROGENEOUS BEDS
A further increase of flow velocity, the bubble rapidly grew to the surface
and formed an eruption from the air inlet to the surface (figure 6.7f-k). The
characteristic V-structure formed, but this time it bended to-wards the right.
This might be explained due to the skewed boundary conditions induced by
the air pocket within the previously extruded surface (figure 6.7e). Segregation between the clay and the glass beads were observed within the fluidized
zone was observed. The clay particles tended to deposit along the boundary
of the fluidized zone (figure 6.7l).
a
b
c
d
e
f
g
h
i
Figure 6.8: Experiment on a layer of glass beads in a packing of clay. A
fracture occured above the inlet at the initiation of the experiment (b) that
propagated into the glass layer (c-e) and pulled clay layer into the glass (e).
Tendency of doming is observed (e). A fracture forms at the edge of the
dome(e) which propagated to the surface (f) forming a surface eruption (g).
Evolvment of fracutre network (h-i) at increasing flow velocity.
6.2.4
Experiments of one layer of glass beads
In this experiments we did the of the previously three experiments. It consisted of a high permeable layer of glass beads within a bed of clay particles
(figure 6.8a). When the experiment started fractures within the lower layer
of clay occurred just above the air inlet (figure 6.8b-d). The fracture propagated into the glass layer and lifted the upper clay section and formed a dome
like structure along the surface (figure 6.8e). Characteristically, a fracture
opens in lower the edge region of the dome due to stretching during doming.
93
CHAPTER 6. ADDITIONAL EXPERIMENTS
This fracture grew and formed an eruption on the surface (figure 6.8f). The
small clay particles was pulled into the glass through the fracture (figure
6.8e-f).
By increasing the flow velocity, the eruption on the surface grew further.
Simultaneously the fracture network developed in the lower clay section (figure 6.8g). The new fracture network made a new large fracture horizontally
above the inlet causing the previously fracture to collapse, and forming a
conduit vertically above the inlet.
6.2.5
Experiment on large glass beads
The system consisted of glass beads of two different sized glass beads, dlarge '
1.8 ± 0.1mm and d ' 630µm and a fill height of h = 14 cm. A picture of the
initial and end state can be seen in figure 6.9. The flow velocity necessary
to fluidize the bed was vf = 54 m/s. This value twice as high as what was
measured in the fluidization experiments in chapter 5. Though one should
be careful by comparing the measurements since they were performed on
different setups. Segregation is observed where the small grains settle along
the boundary of the fluidized zone.
We observe a tendency of the small grains to settle along the boundary
of the fluidized zone. Segregation is a well documented physical process,
but poorly understood. No consise physical argument have been found to
explain the observed dynamics. This tendency is also recognized in kimberlite
structures.
Figure 6.9: Example of segregation phenomena with different sized particles.
The dyed black small beads (d ' 630mum) tend to settle along the boundaries of the fluidized zone and on the surfac close to the conduit.
94
6.3. INTERMEDIATE COHESION
Summary
An observation on the fluidization experiments is that the placing of the
induced heterogeneities (clay layers) did not seem to effect the geometry of
the fluidized zone. The zone marking the transition between the fluidized
and static granular media is plotted in figure 6.10A. By reading off saddle
width can from figure 6.10A we find that the saddle width is about a factor
1.2 times the fill height which is remarkably similar to what we find in the
dry glass bead experiments (figure 5.9A). The experiments plotted are the
fluidized bed of glass beads layered with clay in different regions (in figure
6.5, 6.6, and 6.7) and the experiment with large beads (in figure 6.9).
When the bed is fractured, as in the experiments on wet glass beads
(figure 6.3) and the clay layers (figure 6.2), the fractured zones are highly
irregular. The fractured zone of the wet glass beads and the clay layer are
rescaled and plotted together in figure 6.10A and 6.10B respectively. It can
be seen that the morphology of the zone is highly irregular event though for
identical experiments. This might suggest two things: First is that characterizing the fracturing pattern by plotting the outer zone of the fracture
network is a bad idea. Secondly that the process of fracturing is stochastic
in nature.
6.3
Intermediate cohesion
Experiments have been performed on dry and fully saturated granular materials; little work has been done in the intermediate regime. When a granular
material is partially wet, liquid bridges form between the grains. The surface
tension of the liquid thus provided by these liquid bridges provides potentially an effective way of altering the cohesive force between the grains. As
seen in the chapter 3, granular material with cohesion, differ significantly
in their properties from the dry, cohesion less materials. As we saw, they
can undergo fracturing. A lot of work has been invested at PGP to study
the very interesting transition between fluidization and fracturing in porous
media experimentally. These experiments have so far failed.
The idea was that by varying the humidity of the air by passing it through
a salt solution, one could vary the cohesive forces within the bed. Several
studies (e.g. see Halsey et al [37]) has revealed that humid air induces a liquid
bridges with a given curvarure between particles in a granular packing.
We used a wide range of salts to reduce effect the humidity of the air.
The different salts that we used reduced the vapour pressure of the water
by a factor fs , so pav = fs pv,H2 O . Thus the difference in vapor pressure
95
CHAPTER 6. ADDITIONAL EXPERIMENTS
within the liquid bridge and the passing air can be written ∆pv = pav −
pbv,H2 O = pv,H2 O (1 − f ). Inside the liquid bridge the pressure is reduced
and by balancing the forces around the bead, the net force sticks the beads
together. The cohesive force can found by multiplying the pressure difference
by the area of the liquid bridge Ab ,
Fc = A∆pv
= πRb2 pv,H2 O (1 − f )
→ Fc ∝ f
(6.1)
(6.2)
(6.3)
Thus the cohesive force is proportional to the humidity.
Now the salt solutions did not produce the desired air humidity at the
sufficient rate (∼50 liter/min). I performed experiments at several places
in the experimental circuit all with negative results; the container did not
to produce the desired humidity. This is actually quite reasonable, when
considering the 25 l container containing approximately 3-5 liters of salt
solution and the amount of air passing through the setup. In it self, this could
potentially not be a big problem as long as we could measure the humidity
of the injected air. The two main problems when doing these experiments
will now be presented.
Liquid bridges and humidity - problem 1
How do we know that the equilibrium between the size of the liquid bridges
and the humid air is reached?
One should expect that this equilibrium is reached when the humidity of
the injected air equals the air that has passed through the bed of glass beads.
I did measurements of both the injected air and the air that had passed the
bed. I did this both when small amounts of water was added to the bed,
so the humid air should transport out the excess water, and on a initially
dry bed when the humid air should form liquid bridges at the grain-grain
contacts. I found that equilibrium between the two humidity measurements
were far from reached for experiments up to ∼ 10000 seconds. That suggests
that reaching the equilibrium state between the relative humidity and size of
the liquid bridges is a slow process.
Aging properties of granular media is studied by Restagno et. al. in
2002 [84]. They find that for relatively high humidity the aging effect on the
maximum stability angle θmax increases dramatically with humidity of the
surrounding air in spheres of 200µm in diameter. The key argument I will
use from this article is the time-span of their measurements. They report
a steady monotonic increase of the stability angle up to 105 seconds, i.e.
96
6.3. INTERMEDIATE COHESION
three days. These experiments supports my findings for the time span of
equilibrium between the humid air and the curvature of the liquid bridges.
It is a slow process.
Pressure and relative humidity - problem 2
The dew point of water decreases with the confining pressure. Thus the
relative humidity increases at higher pressure from the definition. Relative
humidity is defined as the reciprocal of the absolute humidity over the dew
point. When we have a pressure gradient across the bed we know that there
is a proportional gradient of relative humidity and thus also the cohesive
forces within the bed. This effect produces cohesional inhomogeneities in
our sample. The pressure dependency in the relative humidity is being used
by Fraysse et. al. in [44] as a method to induce cohesive forces in a rotating
drum.
To sum up on the status, we found that the process of reaching the equilibrium between the humid air and the liquid bridges took days. Secondly,
that this way of doing it from a theoretical point of view induced inhomogeneities that would make the cohesion inconsequential in a phase diagram.
The two problems presented above produced such major problems to our
experimental setup forcing us to give up investigating the transition between
fluidization and fracturing of granular media in this manner.
97
CHAPTER 6. ADDITIONAL EXPERIMENTS
1.5
1.5
Fluidized zone
z(x)/h, [cm]
1
z(x)/h, [cm]
1
Two clay layers
Deep clay layer
Shallow clay layer
Big beads
Static zone
0.5
0.5
0
−1.5
Wet beads − 14 cm
Wet beads − 10 cm
Wet beads − 18 cm
−1
−0.5
0
x/h, [cm]
0.5
1
0
−1.5
1.5
−1
1.4
0
x/h, [cm]
0.5
1
1.5
Clay 12 cm
Clay 18 cm
Clay 13 cm
Clay 11.5 cm
Fractured zone
1.2
−0.5
z(x)/h, [1]
1
0.8
0.6
0.4
Static zone
0.2
0
−1.5
−1
−0.5
0
x/h, [1]
0.5
1
1.5
2
Figure 6.10: Figure A shows the mapped transition between the static and
fluidized zone for glass beads with clay layers in different positions. It can be
seen that the variation of the induced heterogeneities does not seem to affect
the geometry of the fluidized zone. The fractured zone is plotted in figure B
for the three wet experiments, while figure C is the fractured zone for four
of the clay experiments. The fractured zone varies greatly from experiment
to experiment suggesting that fracturing is a stochastic process.
98
Chapter 7
Discussion
Within this chapter we aim at using the fundamental concepts introduced
in the theory chapters to understand the measurements in the experiment
chapter. I will start by discussing the transition between normal Darcy flow
to the formation of the static stable bubble above the inlet. This will be
done by using different models; Griffith fracture criteria, and a discussion
of the transition between laminar and inertial flow where the inertial term
in Navier Stokes equation (v · ∇v) gets more important at high Reynolds
number.
Then a discussion of the linear dependence of the fluidization velocity
vf with respect to height will be given by use of the two hypothesises for
the onset of fluidization. It will be shown that the experimental results will
weaken the first hypothesis compared and give strength to the second one.
The geometrical measurements done of the fluidized zone will be compared
to the kimberlite pipes. Before the chapter will end width a discussion of the
physical effects that are neglected in the previous discussion.
7.1
Onset of bubbling
From the measured phase diagram in figure 5.7 the onset of bubbling happens
at flow velocity vb = 18.4 ± 2.3 m/s for fill heights above ∼ 12 cm. Above
this fill height the bubbling velocity is roughly constant. At fill heights less
than 12 cm no static bubble has been observed in the experiment. This can
be explained by the fact that the flow velocity necessary to fluidize the bed
vf is smaller than vb thus the direct transition between static and fluidized
granular media.
Large fluctuations in the vb measurement are reported. Speculations have
been done that this might be a trace of the packing history and the existence
99
CHAPTER 7. DISCUSSION
(or in some other examples non-existence) of a force chain across the inlet.
The bubble starts off being very small, but the signature in the pressure
versus flow velocity measures is still quite prominent. It is often seen as
a spike in the pressure measurements, which is supported by the collected
pictures. A fundamental observation about the bubble is that it increases
with increasing flow velocity into the bed. It appears to be static and stable,
in the way that it does not evolve as long as the flow velocity is kept constant.
One might ask why this static bubble forms. Two ideas has been put forward: Griffith mode 1 fracturing and the transition between the importance
of viscous and inertial forces. The two ideas are discussed in the following
two sections.
7.1.1
Griffith mode 1 fracture
As described in the theory section, mode 1 fractures form when the fluid
pressure is increased in materials where the cohesive force is larger than the
radius of the Mohr circle. When the small bubble forms above the inlet,
it appears as being a horizontal fracture. The radius of the Mohr circle is
given by the difference between the maximal and minimal stress direction
(Rm = σ1 − σ3 ). The way of measuring the cohesion within a granular
packing was discussed in section 3.2.1. Remember that Morgues et. al. 2003
proposed that the cohesive forces in a granular packing were a lot smaller,
as low as C '50 Pa, when considering the wall effects [52]. The difference
between the maximal and minimal direction of stress (σ1 − σ3 ) is larger than
the cohesive force (i.e. RM > C). I therefore conclude that mode 1 fracturing
is out of the question when trying to understand the formation of the bubble.
7.1.2
Transition from laminar to turbulent flow
In section 4.3 the concept of Reynolds number marks the transition from the
dominance of viscous to inertial forces. The Reynolds number was defined as
Re ≡ vlν , where v is the flow velocity, l is a characteristic length1 , and ν is the
dynamic viscosity. For low Reynolds numbers (low velocities), Stokes found
the drag force FD to increase linearly with velocity for a spherical particle
moving through a viscous fluid with (see equation 4.29).
At high Reynolds numbers (high velocities), Euler calculated the drag
force on an obstacle moving “quickly” through a viscous media to increase
with the velocity squared. This effect is often seen as a marked transition
between laminar and turbulent flow and the formation of eddies behind the
1
In a tube it is given to be the radius.
100
7.1. ONSET OF BUBBLING
moving object. The transition between the linear and squared behaviour can
be identified by the critical Reynolds number, as is experimentally shown by
Tritton et. al. 1988 in figure 4.3.
It is proposed that the bubble forms due to the change in velocitydependence of the drag force locally as the Reynolds number increases. By
increasing the flow velocity through the inlet into the cell we increase Reynolds number. The critical Reynolds number, marking the onset of squared
velocity dependence in the pressure difference through a pipe is given by
Tritton et. al. to be Recrit ' 3000. Solving the Reynolds number for flow
velocity yields,
Recrit µ
Recrit µ
= √
vb,Rec =
,
(7.1)
ρl
ρ Ainl φ
where the characteristic length l right above the inlet is assumed to be the
square root of the area of flow above the inlet. By equating in values we get,
the onset of bubbling from these set of assumptions is vb,Rec ' 20 m/s which
surprisingly close to what is measured to be the onset of bubbling.
When a force chain passes through the bed above the inlet it is expected
that higher drag force is needed to form the bubble. This suggests that
one should expect large fluctuations in the bubbling velocity dependent of
the existence and or non existence of force chains. This is exactly what we
observe.
Mass of fluidized zone
The total mass of the fluidized zone m can be found from the functional form
of how the transition depends on the height z(x). Since m = φρbAfl , where
Afl is the area of the fluidized zone, b is the distance between the glass plates,
and φ is the porosity. The mass can be found from
m = φρb hW −
Z
2W
0
!
dxz(x) ,
(7.2)
where h is the fill height, W is the width of the activated zone at the surface.
By discretizing the integral and setting dx = W/N ,
m = φρb hW −
N
X
i
W
hi
N
N
1 X
= φρbW h −
hi
N i
!
havg
= φρbW h 1 −
,
h
101
!
!
(7.3)
(7.4)
(7.5)
CHAPTER 7. DISCUSSION
we obtain a simple expression for the overall mass. Since the fluidized zone
scales linearly with height (see figure 5.8), we find the width to scale as
W (h) = 0.8h. Combining W , the average height which is measured to be
havg = 0.5h in equation 7.5, and the porosity φ =0.385, we obtain
m ' 0.4φρbh2 .
7.2
(7.6)
Onset of fluidization
We aim in this section at deriving an analytical expression for how the fluidization velocity vf depends on the fill height. From the measured phase
diagram in figure 5.7 we know that the fluidization velocity increases linearly
with height. The slope in the phase diagram is measured to be af =167±7s− 1.
In the pressure measurements the fluidization occurs when the pressure
between the inlet and surface rapidly drops. The discontinuous drop in the
p(v) is a signature of a first order phase transition, i.e. from static to fluidized granular media. In the fluidized state a high permeable conduit forms
between the inlet and the surface. In the Hele Shaw-cell the fluidization is
seen when the previously static stable bubble rapidly grows to the surface.
The light bubble can grow to the surface when the matrix above behaves as
a liquid. It will in this and the coming sections be determined what causes
the onset of fluidization, whether it is a balance of pressure or due to viscous
drag. I will discuss hypothesis 1 in this section and use hypothesis 2 in a later
section after calculating how the effective permeability changes with height.
The theory presented in section 4.10, where the criteria for fluidization is
when the pressure at depth equals the lithostatic pressure of the overlaying
sediments (hypothesis 1). I will apply this statement to the experimental
setting, thus writing
pf Ainl = Fg ,
(7.7)
where pf is the pressure at the inlet needed to fluidize the bed, Ainl is the
area of the inlet and Fg is the gravitational pull of the fluidized zone above
the inlet. By using Darcy’s law for the pressure, and Newton 2.law for2 the
gravity term, we can solve for the fluidization velocity
vf =
mgkeff
,
hµAinl
2
(7.8)
Sir Isaac Newton (1642 - 1727) was an English mathematician, physicist, astronomer,
alchemist, chemist, inventor, and natural philosopher who is generally regarded as one of
the most influential scientists and mathematicians in history. He wrote the Philosophiae
Naturalis Principia Mathematica in 1687 in which he described universal gravitation and
the three laws of motion, laying the groundwork for classical mechanics.
102
7.2. ONSET OF FLUIDIZATION
where keff is the effective permeability of the bed as it was measured it in
section 5.3.1. A derivation on how the effective permeability in this geometrical setup relates to the permeability measured in a “standard geometry”
(as described in section 4.9) is given later in section 7.3. Using the empirical
expression for the mass of the fluid zone developed in section 7.1.2, in the
previous equation yields,
vf = 0.4
ρgφkeff
h = aanal
h.
f
µAinl
(7.9)
We now have developed an expression for at what imposed velocities to expect
the fluidization to occur as a function of the height. This relation needs to
be compared to the measured value from the phase diagram by comparing
the slopes ameas
and aanal
. By plugging in the given values for ρ, µ, and g
f
f
and the measured values for Ainl , φ, keff , we get
aanal
' (180 ± 20) · 103 s− 1.
f
(7.10)
This is off the measured value of the slope by a factor of 1000. It seems some
values are largely over estimated, for example the assumption that the whole
mass of the fluidized zone is lifted or that the area of the pressure source is
too small. Or we might conclude that the first fluidization hypothesis is fully
flawed. The bed seems to fluidize a lot earlier than when the fluid pressure
equals the lithostatic weight.
The only geometrical consideration done in the presented derivation was
of the mass of the fluidized zone. In the proceeding sections I will approach
the fluidization velocity at the inlet by asking what the velocity at the inlet is
when the top layer of beads lift. Without John Wheelers 1’st moral principle,
“Never calculate anything you do not know the answer for!”
we would happily live on with the calculation above. Since the answer for
vf (h) is known from the measurements a second attempt of the fluidization
velocity will be given in section 7.4. But before that, a few geometrical
considerations of the flow field will be given.
Thus the first hypothesis that fluidization occurs when the pressure at
depth equals the lithostatic pressure in granular media is fully flawed. The
reason for that might be related to the following. Due to the nature of granular media causing e.g. large spatial fluctuations in the stress measurements
on the bottom of a container is the lithostatic pressure not well defined.
These fluctuations are thought to exist due to the formation of force networks within the media. This misconception rises from the thought of rocks
being viscous, which they definitely are on long time scales.
103
CHAPTER 7. DISCUSSION
7.3
Calculated effective permeability
Now the measured permeability, or the effective permeability keff , in the
experimental setup will due to the geometry of the setup deviate from the
permeability as a material property given by the Carman-Kozeny relation.
Two separate reasons can explain this
• the packing of beads is reduced along the walls hence increasing the
effective permeability. In the experimental setup the width of the cell
is 0.8 cm and the average diameter of the beads are a ' 630µm. Thus
the wall effect can maximum change the porosity by a factor fw = 630
µm/0.8 cm = 0.08 or 8%.
• the geometry of the setup yields another way of measuring the permeability than what is usual. Normally the flow cross sectional area is
kept constant through the flow, but here the inlet area is much smaller
than the surface area by a factor f = Asurf /Ainl .
In the following I will calculate the effective permeability keff by considering
the geometrical property of the setup. Thus the wall effect on bulk permeability is neglected.
Now the area available for flow is given by the distance between the plates
(0.8 cm) times the red line in figure 7.1.
Figure 7.1: The red line shows the assumed linear dependence of the width
x as a function z. The surface area is given by as and the inlet area is given
to be ai .
104
7.3. CALCULATED EFFECTIVE PERMEABILITY
Through mass conservation for incompressible fluids, the volume flux into
the bed equals the volume flux out through the surface
(7.11)
(7.12)
(7.13)
φinl = φsurf
→ v i ai = v s as
= v(z)x
Where v is the volume flux per unit area i.e. the flow velocity, vi is the inlet
velocity, and vs is the surface velocity. Now in two dimensions the length x
available to flow is assumed increases linearly with z (see figure 7.1) through,
z
x(z) = ai + (as − ai ),
h
(7.14)
where ai is the inlet area, as is the effective surface for the air flow of the bed,
and h is the fill height. We continue by setting fA = as /ai . Using equation
7.14 in 7.13, the fluid velocity is given to be
v(z) =
1+
vi
.
(fA − 1)
z
h
(7.15)
In combination with Darcy’s law 4.37, we find obtain a differential equation
for the pressure
vi
kck ∂p
=−
.
(7.16)
z
1 + h (fA − 1)
µ ∂z
By substituting u = 1 +
z
h
(fA − 1) and ∂u =
fA
∂z,
h
we obtain
∂p
µvi h
=−
.
∂u
kck fA u
(7.17)
This differential equation is separable and can be solved by integrating both
sides yielding a solution for the pressure where we have substituted back the
z
z
µvi h
ln 1 + (fA − 1) + A1 .
(7.18)
p(z) = −
kck fA
h
We can find the integration constant A1 , which we know is independent of
both p and u by noting that p(0) = A1 = pi , i.e. the integration constant is
the pressure at the inlet. Now
p(z) = pi −
µvi h
z
ln 1 + (fA − 1)
fA
h
we see that the pressure is reduces by the logarithm of z.
105
(7.19)
CHAPTER 7. DISCUSSION
Now if we equate the theoretical prediction of the pressure difference ∆p
across the bed and the measured pressure difference ∆p0 ,
p(0) − p(h) = p0i − p0s
hµ
µvi h
ln [fA ] =
vi
⇒ p0 − p0 +
kck fA
keff
Solving this equation for the effective permeability
fA
keff = kck
.
ln(fA )
(7.20)
(7.21)
(7.22)
fA
We see that the effective permeability increases by a factor ln(f
from the
A)
Carman-Kozeny relation for the permeability kck as a material property. This
is a transcendent equation that has no analytical solutions for fA 3 . Where
fA = as /ai = c2d h/ai , where c2d as defined as how the area of the surface
depends on the fill height.
∂p
With the measurements of the slope a ≡ ∂v
given, the effective permeability can be found from Darcy’s law through,
µh
.
(7.23)
keff =
a
The effective permeability is plotted for all fill heights in figure 7.2.
The effective permeability is found to vary from keff = (0.4 → 1.8) · 10−8
m2 . With the Carman-Kozeny relation for the permeability given to be
1.33·10−9 m2 , the factor fA can be found from the transcendent equation 7.22
relating the effective and the Carman-Kozeny relation for the permeability.
The equation
fA
= 0,
(7.24)
keff − k
ln(fA )
was solved numerically for fA . The factor relating the inlet and the surface
area was found to be fA = (201 ± 34)h.
We have now developed a semi analytical expression for the Darcy slope
µ
µh
=
ln(Cg h)
(7.25)
a(h) =
keff
Cg k
µ
a(h) =
ln((201 ± 34)m−1 · h)
(7.26)
(201 ± 34)m−1 · k
A logarithmic dependence between the Darcy slope and the fill height is thus
analytically derived.
By plotting the analytical solution of a(h) and the measurements of a in
figure 7.3, the logarithmic dependence of the slope is reproduced.
3
The analytical solution for fA can thus be found by using the Lambert W function
defined by the inverse function of f (W ) = W eW .
106
7.4. ONSET OF FLUIDIZATION, 2. ATTEMPT
−8
2
x 10
1.8
Measured permeabilities
Calculated permeability with f known
A
Calculated permeability ± σ of f
1
A
Permeability, [m2]
1.6
1.4
1.2
1
0.8
0.6
0.4
0.05
0.1
0.15
Fill height, [m]
0.2
0.25
Figure 7.2: This is a plot of the measured permeability versus fill height.
The effective permeability is found to increase almost linearly with fill height,
but with a slight bend at large fill heights. This suggests almost no height
dependency in the slope between the pressure and velocity measurements. It
is observed that the effective permeability ranges between 0.4 → 1.8 · 10−8 .
7.4
Onset of fluidization, 2. attempt
In the previous section we derived an expression for how the effective, measured permeability related to the Carman-Kozeny relation for the permeability
when measuring it in a normal setting where the cross sectional area is kept
constant. We found that the effective permeability was given by the “true”
fA
where fA = as /ai . In section 4.10.1 we
permeability times a factor ln(f
A)
calculated that a flow velocity of 1.3 m/s would lift a single bead with no
load on top. When considering a bed of beads this value would potentially
be a bit lower due to neighbouring effects.
Through conservation of mass, we have that the volume flow through
the surface b equals the volume flow through a (hence assuming compressible
fluids). Through the factor fA we can now find the imposed velocity necessary
to fluidize the bed vf (h) through the inlet ai , as
va = vf (h) = fA vsurf .
(7.27)
Now as previously derived, fA = (201 ± 34) m−1 h, thus
vf (h) = (201 ± 34)m−1 h · 1.8m/s
107
(7.28)
CHAPTER 7. DISCUSSION
Measured slopes and analytical solution
320
300
Measured slopes in the linear regime
Analytical solution for the slope
σ1 contours
280
dp/dv, [Ns/m3]
260
240
220
200
180
160
140
0.05
0.1
0.15
Fill height, [m]
0.2
0.25
0.3
Figure 7.3: A plot of the measured slope in the linear regime between the
pressure drop and flow velocity is plotted together with the analytical solution. It can be seen that the analytical solution almost reproduces the measured values.
= (361 ± 61)s−1 · h.
(7.29)
We have now derived an expression for the imposed fluidization velocity necessary to fluidize the top layer of the bed. In figure 7.4 the analytical solution
for the fluidization transition and bubbling transition is plotted together with
the measurements and the linear best fits. It is observed that the transition
is successfully reproduced.
The bubble rapidly grows to the surface due to buoyancy effects when the
weight of the overburden is liquified, i.e. when their weight is counteracted
by the viscous drag of the fluid.
7.5
Natural systems
Htvc and kimberlites shows evidence of fluidization of brecciated elements [6],
[4], and [3]. The fluidization is thought to be induced by fluid flow caused
by high fluid pressures at depth. Striking similarities are observed when
comparing the cross sections of the three dimensional objects in geology
(figure 2.3) and the fluidized zone in laboratory (figure 5.6). A well defined
fluidized zone develops that consists of two parts; a steep angle (α2 ) lower
108
7.5. NATURAL SYSTEMS
120
100
Measured vf
Measured v
b
Linear best fitted vf
Average vb
Calculated vb
Calculated vf(h)
σ for calculated v (h)
Flow velocity, [m/s]
1
f
80
60
40
20
0
0.05
0.1
0.15
0.2
Fill height, [m]
0.25
0.3
Figure 7.4: This is a plot of the measurements and the analytical solution
for the fluidization velocity and bubble velocity. The bubble velocity is well
reproduced by the analytical solution for vb . The analytical solution for the
fluidization velocity is about 100% higher than what was measured.
pipe zone and a more shallow angle (α3 ) crater zone. The two zones are
observed in both class 1 and class 3 kimberlite pipes [6]. The angle (α2 ) of
the pipe wall is in the experiments found to increase from approximately 60o
to 70o for fill heights h from 5 to 25 cm (figure 5.9f). The values of the angles
are comparable to the pipe wall angles for kimberlites. Class 1 kimberlites
set in are steep with an angle of 75-85o, while class 3 can be as shallow as
∼ 45o positively dependent on the level of consolidation of the local geology
[6].
There is one additional feature seen in the experiments that look similar
to what is documented in geology. Inward dipping of high reflector layers
around htvc are documented by several authors e.g. in the Vøring- and Møre
basin by Svensen et. al. in 2004 [9]. This is also observed in the laboratory
(see figure 5.5 along the margin of the fluidized zone where the grains flow
downwards. This might be interpreted as a signature of similarities of the
flow pattern of the brecciated elements (granular media) in the htvc. In the
the htvc fully developed fluidization is not expected after the initial burst.
The cartoons of the flow patterns within kimberlite structures are based on
viscous induced fluidization experiments e.g. [3] and [6], similar to mine. My
experiments (see figure 5.5) reproduce these patterns as drawn by Walters
et. al. in 2006 [6].
109
CHAPTER 7. DISCUSSION
When two objects on such a different scale look this similar it is tempting
to propose that they formed with the same process. This might be a bit to
harsh statement due to the following arguments.
How to understand?
All natural sciences are build on observations of the nature. Several sciences
have had problems of exceeding this level. This partly applies to sciences
such as geology and biology. Purely reporting of observations without using
them to shed light on universal relations and laws of nature is intellectually
unsatisfactory. However, some research communities attempt to exceed the
level of “stamp collection”.
So what does it mean to understand your observations? The only way
of understanding, is in my point of view, by developing testable analytical,
numerical or analogue models. The study of geology is mainly a study of
fossil structures where both the initial conditions and the process are more
or less unknown. No one was there when the forming process took place.
When our model reproduces the observed structures our model is supported.
However, one would never for sure know whether the model captures the
right process, nor do we know if we had the correct initial conditions. In
principle there are an infinite number of models, and even worse, an infinite
number of initial conditions that can reproduce the end product, i.e. the
fossil feature we observe today. The question of choosing between models is
a difficult question. An often used strategy of chosing is to apply Occams
Razor, as nicely put it by R. Dunbar in his book from 2004;
“There is little to be gained by having an explanation that is so
complex or difficult to confirm that we waste valuable time on it
when we could be out foraging or finding mates.”
-Robin Dunbar [85]
To end at a conclusion is difficult, though I claim that to understand
geological features one needs to study the fundamental physical processes.
In the next section I will be discussing how two dimensional studies can
increase the understanding of three dimensional objects.
7.5.1
2D versus 3D modelling
I will now give a qualitative discussion of how the quasi two dimensional
geometry of the Hele-Shaw cell can or can not relate to the three dimensional
geological structures.
110
7.5. NATURAL SYSTEMS
The geological structures (htvc and kimberlites) are circular, axis symmetric, cone shaped structures formed in a 3 dimensional setting. In the
experimental setup we have the formation of a planar 2 dimensional structure. We therefore ask the question whether the process of fluidization forming the feature is independent of the number of dimensions. If the balance
of the forces depends on the number of spatial dimensions, the qualitative
discussion of the fluidization may potentially flaw.
Dimensionality problems arise in elastic models. In a 1D elastic model
the only governing physical parameter is the Youngs modulus, while in 2D we
have both the Youngs modulus and the Poisson ratio. So in a 1D elastic model
we would miss important physical processes in objects more complex than
rubber bands. We would thus not qualitatively reproduce the true natural
behaviour of an elastic body where the area of the orthogonal direction would
decrease. No new physical effects would come into play by increasing to three
dimensions, thus a 2D elastic model would qualitatively reproduce the 3D
process. However, the quantitative predictions could potentially deviate from
the true value.
A second relevant example is viscous flow in porous media which is governed by Darcy’s law. This expression is independent of the number of spatial
dimensions and no new physical quantities occur when increasing the number
of dimensions.
As we have seen from the two examples, there are potential some problems
in the application of lower dimensional models. These problems rise when,
(1) the forces in a force balance equation scales differently with the number
of dimensions and (2) when complexities and physical effects such as Poisson
ratio disappear in lower dimensions.
In the presented study equations governing gravity, Darcy’s law of porous
flow, viscous drag on sedimenting particles and pressure are all independent
of the number of dimensions. Gravity and Darcy’s law are the same in all
dimensions, though the viscous drag of a sedimenting particle only applies
to two or more dimensions. The pressure is dimension less, so I do not miss
any processes and physical parameters in my 2D study that exists in the 3D
applications.
However, some differences exist due to the geometrical difference between
the experiment and geological objects of interest. These geometrical differences do not seem to be related to the processes as described in the previous
paragraph, but due to the boundary conditions. The walls of the Hele-Shaw
cell sets definite limits of the formation of the fluidized zone. In stead of
making 3D cone shaped objects we have a quazi two dimensional object that
looks remarkably similar to a cross section through the centre of the geological structures. In addition are there friction between the walls of the
111
CHAPTER 7. DISCUSSION
container that is not occurring in geology. For these reasons, will the scaling
of the quantitative measurements in the experiment potentially deviate from
what is expected in a 3D setting. To improve the predictions of flow localization and fluidization one needs to build a setup that better approach the
boundary conditions in geology.
2D model is a good first approximation to quantify the weight and interaction of the different processes. It is also nice visually. One obtaina real
time images of the how the different forces interact within the packing. However, a 3D model is necessary to get applicable quantitative results in order
to fully understand the 3D vent structures.
7.5.2
Piercement structures in nature
In this section I will discuss the flow velocity diagram as presented it in
figure 5.12 followed by my interpretation of how piercement structures form
in natural systems.
Now fluidization occurs when the viscous drag of a fluid equals the gravity.
In figure 5.12 we calculate the fluidization velocity when vary the depth of
the fluidized zone from laboratory to kimberlite scale (i.e. from 1 m to 100
km) along the x-axis. Along the y-axis we vary the permeability from granite
to gravel (i.e. from 10−20 m2 to 10−10 m2 ).
In nature there are a wide range of fluids that induce the fluidization, from
magma in kimberlites to gas and water in htvc. The physical properties used
in the dimensional analysis for the liquid is the compressibility and viscosity.
In the diagram we plot the fluidization velocity for these three liquids in
the horizontal direction. What we find is that magma has the lowest flow
velocity neccessary to fluidize the matrix. The fluidization velocity increases
two orders of magnitude when going from magma to air, and two orders again
when going from air to water.
The host rock property will also vary. This is done vertically. I assume
that the density difference between the viscous liquid and host rock is the
same for all three cases. The density difference between the magma and gas
is of course acknowledged, despite of that I assume it to be inconsequential
to the process. In this model I vary the cohesion of the rock from 0.1-10
MPa. By increasing the cohesion by one order of magnitude (downwards)
the flow velocity increases two orders of magnitude.
Interpretation of piercement formation
The discussion is based on the intuition that is build during discussions and
the laboratory work. Several others have previously discussed the formation
112
7.6. ADDITIONAL PHYSICAL EFFECTS
of piercement structures in htvc and kimberlite settings. Two excellent reviews are Planke et. al. 2003 [7] and Sparks et. al. 2006 [86] respectively.
(1) When the fluid pressure (for different reasons explained in chapter 2)
increases fracturing patterns and brecciation develops in a zone above the
cause of the pressure build up. The fluid flow reaches near surface through
the brecciated zone. By near surface in the htvc case, I may refer to the
lower part of the eye-structure observed in seismic in figure 2.4, and for the
kimberlite the root. (2) Here it starts disintegrating into explosive flows that
fluidizes (3) the overlaying brecciated elements. The fluidization occurs when
the viscous drag of the fluids equals the gravitational weight of the overlaying
bed. Field evidence from Karoo in South Africa shows the occurrence of in
situ brecciation (pers. com. H. Svensen). It is the remains of the fluidized
zone of the brecciated elements that we today see as the pipelike structures.
I. e. the pipes are a trace of fluidization.
7.6
Additional physical effects
Several additional physical effects have arised in the experiment that is neglected in the interpretation and discussion of the onset of fluidization. Some
of these effects might have quantitative effect on the calculations. Now the
calculated value for the slope of the fluidization velocity versus height is about
a factor two higher than what was measured. Thus the onset of fluidization
happends at a lower fluid velocity. We will now go through the effects that
have been neglected in the interpretation.
Bending of the glass plates
When fluid pressure is induced to the bed, some bending of the glass plates
are expected. The elastic modulus of glass is in the order of ∼ 100 GPa [39]
and the fluid pressure is in maximum in the order of 10·104 Pa. Now the stress
is given by Youngs modulus times the strain, so the strain ε = σ/E = pf /E,
where the engineering strain is given by ε = ∆l/l. This yields an expression
for the displacement
pf
∆l = l ∼ 10− 5m.
(7.30)
E
The value for the bending is three orders of magnitude less than the width
of the cell, and about 1/50 of the grain diameter. Bending of the glass plates
as an extra physical effect can thus be ruled out of the discussion.
113
CHAPTER 7. DISCUSSION
Thermal fluctuations
Thermal fluctuations within the measurements are suggested to affect the
experiment. For dynamic granular media the temperature is of no importance
since the thermal energy is a lot lower than the gravitational energy (kT mgd). Since temperature fluctuations can alter the relative humidity, the
cohesive force within the bead could be altered. In the chapter 6 experiments
of altering the cohesion is discussed. A major problem with those experiments
were the long time (∼ 10·4 s) needed for the liquid bridges that would induce
the inter particle cohesion to form. It is speculated that the temperature
effects the surface tension of the water and thus the cohesion wihin the bed.
Temperature measurements between 21 − 24o C were performed, it did not
seem to effect the fluidization velocity.
Elasticity of the hose
The hose that supplied the air was made of plastic and could thus yield
elastically to the imposed air pressure. Higher fluid pressure would increase
the radius the hose thus alter the flow velocity calculated at the inlet. An
increase of the radius of 0.2 mm would decrease the flow velocity by ' 1%
(from equation 5.5). The elasticity of the hose is thus neglected.
Static electricity
Static electricity has been proposed to affect the dynamics due to the observations that single beads would hang on the walls (see figure 5.6). Even
though that the static electricity might be important on grain level, no such
dependence seems to be important in the bulk. This statement is based on
observations in the laboratory and the lack of this dependency in the physics
literature.
Flow feed back on porosity
Now the grains that we use in the experiments are Type B-particles, as
defined by Geldart in 1973 [61]. The classification is partly based on the
observation that the porosity of the bed increases with the air flow. No
such feed back mechanism has been taken into consideration when doing the
calculation of the fluidization velocity. An increase of porosity (thus also
permeability) would from the dimensional analysis increase the flow velocity
necessary to fluidize the bed (figure 5.12). When neglecting this effect the
expected fluidization velocity should be lower than what was measured.
114
7.6. ADDITIONAL PHYSICAL EFFECTS
Reynolds found that granular materials had to dilate to accommodate
any shear. This effect can be seen in figure 7.5.
Figure 7.5: In this figure we see how the air flow through the media effects
the porosity and thus the permeability of a zone above the inlet. The more
transparent the bed, the higher porosity, and thus permeability (see CarmanKozeny relation 4.58). It can also be seen from the figure that the height
from the top of the bubble to the surface h0 is smaller than the initial fill
height h. From Darcy’s law we know that an increase of permeability and
decrease of fill height reduces the pressure difference between the inlet and
the surface.
Isocontours of the velocity field
In the calculation for the flow velocity it was assumed that all particles fluidized simultaneously within the surface area as . Though due to the fact that
the isocontours of the flow velocity is larger in the centre along the surface
(see figure 5.14) it is expected that grains would fluidize in the centre above
the inlet before on the sides. Thus one should expect that the grains in the
center would reach the fluidization criteria at lower imposed fluid velocities. It is speculated that this effect can potentially reduce the surface area
that fluidizes by a factor two thus saving the calculation for the fluidization
velocity.
115
CHAPTER 7. DISCUSSION
Summary of the discussion of extra physical effects
In this chapter several additional physical effects were discussed. When
back-of-envelope calculations or measurements could be made thermal fluctuations, elasticity of the hose, bending of the glass plate, and to some extent
static electricity were ruled out. In essence what is left with that could potentially reduce the value of the analytical expression of the fluidization velocity
by using the isocontours of the flow field. When the isocontour of the necessary flow velocity touches the surface (in the centre), the bubble can grow
to the surface due to buoyancy. A second effect that would increase the
necessary flow velocity, is the fluid velocity feed back on the porosity. These
three effects should be considered when calculating the fluidization velocity.
116
Part IV
Concluding Remarks
117
Chapter 8
Brief summary and conclusions
An experimental setup was developed to study phase transitions in granular media. This setup was developed by Sean Hutton at PGP during 2003.
Within the setup we can control the flow velocity and measure the pressure
difference thus revealing the bulk behaviour in the bed. By varying the fill
height and controlling the flow velocity into the Hele Shaw cell a phase diagram of the documented feature was developed. The phase transitions were
recognized by a rapid change in the bulk behaviour of the bed in comparison
with image analysis.
Transitions between static and fluidized granular media with an interconnected bubbling regime (for fill heights above 12 cm) was observed. It was
found that the flow velocity necessary to fluidize the bed increased linearly,
while the flow velocity making the bubble is independent of fill height.
The bubble is interpreted to occur as a shift from a linear to squared
dependency of the flow velocity in the drag force above the inlet. By using
the critical Reynolds number from Tritton et. al. we find that the bubbling
fluid velocity to be ∼ 20 m/s which reproduces the measurements.
The bubble rapidly grows to the surface when the overlaying sediments
are lifted by the viscous drag from the air flow given by Stokes relation for a
sedimenting particle, i.e. when the granular media liquefy. The flow velocity
at the inlet related to the surface velocity through vinl = fA vsurf = aasi vsurf ,
from mass flow conservation. By taking the geometry into consideration fA
was found from the transcendent equation relating the effective measured
permeability kef f to the “true” Carman Kozeny relation, thorough
kef f = kCK
f
h
∝
.
ln(f )
ln(h)
(8.1)
The linear over logarithmic height dependency of kef f lead to a logarithmic
height dependency of the slope a in the linear regime of the pressure versus
119
CHAPTER 8. BRIEF SUMMARY AND CONCLUSIONS
velocity measurement. This logarithmic dependency is of a and kef f is supported by the measurements.
Through dimensional analysis and with the functional form of the fluidization velocity f (α1 ) given, quantitative predictions of fluidization and flow
localization in naturals systems are given.
We find that hypothesis 1, that fluidization occurs when the pressure at
depth equals the lithostatic pressure of the overlaying sediments, is strongly
weaken. The experimental measurements supports hypothesis 2, that fluidization occurs when the viscous drag of the imposed air flow equals the
gravitational force.
Several additional experiments are performed. Experiments on clay particles,
mixture of clay and glass beads, and wet glass beads were performed. By
adding water to the glass beads, inter particle cohesive forces are induced to
the bed. When the flow velocity was increased, fracturing in stead of fluidization is observed in the higher cohesional media. The competition between
fracturing and fluidization is very interesting process. Quantifying the transition between fluidization and fracturing behaviour in granular media would
be very interesting.
The geometry of the fluidized zone seems to be independent of the material used when fluidization is the prominent physical process occurring within
the bed. When fracturing, the geometry of the fractured zone differs significantly from experiment to experiment and when changing the material. This
suggests that fracturing is a process more stochastic in nature.
The main conclusion of this thesis is that fluidization occurs when the
viscous drag equals the weight of the overlaying sediments, i.e. hypothesis
2 is supported. That insight builds the intuition for when granular media
behaves liquid like. The presented conclusion counteracts what is referred to
as the conditions for venting as it is given by several authors of this topic in
the htvc literature [5], [7] and [9]. However, in the kimberlite literature the
supports the presented criteria for fluidization e.g. [6].
Non dimensional ratios are developed and in combination of the functional
form of f (α1 ) measured in laboratory it enables us to quantify under which
conditions fluidization of brecciated elements (granular media) may occur in
geology. A contour plot of the neccessary fluidization velocity is given when
varying µKb for the liquid, C and k for the host rock and h for the depth of
the root of the fluidized zone (figure 5.12). We have also given a plot of when
to three different fluid flow velocities when varying only the depth and permeability based on the same dimensional analysis and measured functional
form of f (α1 ).
120
Chapter 9
Future work
Through the experiments and the theoretical discussion a deeper understanding of the physical process occurring in the cell is achieved. A success story
has been produced by linking the theoretical considerations to the measured
observations. We miss by only a factor two in the slope.
Several assumptions are made along the way so we do not claim that
the presented theory of fluidization is the full story. In the future, there are
several aspects of the theory and experiments that could be improved. The
importance of the extra physical effects should be tested to strengthen (or
weaken) the arguments used in this thesis. In the following I will present
what I feel is the most important physical effects might occurring in the bed
to be explored.
Wall effect
It has been suggested that the formation of a static stable bubble occurs due
to wall effects. Janssen derived an expression for the wall effect as described
in the introduction of the thesis. He found that an increasing proportion of
the weight was hanging onto the walls of the cylindrical container when the
fill height was increased. This relation has later been experimentally tested
and found to be important of measurements of the vertical stress at the
bottom of containers. Undoubtedly this effect is also occurring within this
experimental setup. The importance of this effect depends on the frictional
coefficient between the glass plates and particles within the bed and the
geometry of the container.
The importance of this effect can quite easily be tested within the same
setup by slight modifications of the Hele Shaw cell. By detaching the bottom
of the Hele Shaw cell one can measure the weight of the bed bed when varying
the fill height. These measurements will reveal whether the wall effect plays
121
CHAPTER 9. FUTURE WORK
any significant role on the stress field on the bottom of the cell.
It was quite early decided that the work should focus on doing several
measurements on the phase diagram before the setup was modified. Due
to time constrains and the fact that the Hele Shaw cell will be used in the
future, this has not yet been done.
Flow velocity feed back on porosity
It is observed that before fluidization a zone above the inlet brightens. The
more light reaching the camera, the higher porosity is expected within the
bed. Reynolds described that due to geometrical considerations within the
packing structure that the porosity should increase for the bed to accommodate any shear. The flow dependency of the porosity is neglected in the
theoretical discussion.
By measuring the light intensity and the volume expansion due to dilatancy,
the increased porosity can be related to the flow velocity since the number
of grains is conserved and the flow velocity is imposed.
Measurements of the grain size distribution
When calculating e.g. the Carman-Kozeny relation for permeability, the
average grain size is used. The producer of the glass beads have not been
able to give us the grain size distribution thus the average grain size is in
practice unknown.
The size distribution can quite easily be measured by programs developed
by e.g. my fellow master students. Due to their stress of finishing their master
thesis, this has not been done yet. Once the grain size distribution is known
the average grain diameter can be calculated.
Fluid flow velocity above maturated aureoles
To better constrain the fluidization criteria in natural settings, the flow velocities thorough sediments above maturating aureoles is needed. When the
aureoles are heated, phase transitions within the organic material will rapidly
increase the unit volume of the gas. The rate at which the unit volume of the
organic material at depth increases, will determine the flow rate through an
imagined horizontal surface above the aureole. In an infinitely large medium,
the gas expansion will only occur toward the free surface above. By calculating the rate of the volume expansion from thermodynamic and chemical
arguments the flow velocity can be found.
The rate at which the aureoles are heated is given by the thermal diffusivity of the medium plus the temperature of the gas flowing upwards. The hot
122
gas flow would increase the rate at which heat is transported to overlaying
sediments, thus a feed back on the volume expansion rate is suspected.
Once the flow velocity in natural processes are found, a model is in this
thesis developed to quantify under which conditions venting occurs when
permeability is plotted against depth. A numerical model taking this process
into consideration will be developed by Ingrid Aarnes in her PhD degree.
3-D setup
Due to the geometrical constrains in the Hele Shaw cell it would be very
interesting to develop a similar phase diagram in a three dimensional setting.
These measurements would improve the quantitaitve geological applications.
It could also be interesting from a physics point of view to quantitatively
compare how the geometry of the cell affects the results.
A large container (1x1x1 m) is build that would be a great tool to use
in the formation of this phase diagram. Within the same container, the size
of the inlet can be varied. In geology the sills heating the aureoles are up
to 20 square kilometres. Thus the point source, as in the laboratory, is not
occurring in geology.
On this container the question of how the geometry of the inlet or the
container it self effects the fluidization velocity can be revealed.
Fracturing versus fluidization
As has been previously described, the competition between fracturing and
fluidization of granular media is of major importance. A fundamental observation is that Mode 1 fractures occur in highly cohesive media, while fluidization occurs in cohesion less media. A second observation it is energetically
easier to fracture a cohesive media than fluidize a cohesion less media.
I speculate that fluidization as a process occurs in media where the force
ratio of gravity over cohesion is very large, and vice versa. So by decreasing
the grain size and keeping the cohesion constant, or vice versa by keeping
the grain size constant and increasing the interparticle cohesion, we would be
able to experimentally examine the transition from fluidization to fracturing
behaviour.
These experiments can be done in several ways. Forstyh et. al. [38] has
induced the cohesion by inducing a magnetic filed to the bed. This can in my
opinion be nice way of controlling the cohesion. Halsey et. al. [37] showed
that the cohesive force is a function of the size of the inter particle liquid
bridges within the bed. By assuming that the size of the liquid bridges is
dependent of the relative humidity of the air that passes through the bed,
123
CHAPTER 9. FUTURE WORK
the cohesive force can thus in principle be varied. Several problems by doing
this have previously been discussed. As a short re cap for the sake of the
argument is that reaching the equilibrium between the liquid bridges and the
humidity is extremely slow process, and homogeneously control the relative
humidity through the sample is difficult. It is therefore suggested that the
cohesive force should be prepared by other means.
B. Phillips will do his master thesis by varying the cohesion by draining
a water-vanish mixture through the porous bed. When draining the sample
over night, the varnish glues the beads together inducing inter particle cohesive forces. By varying the varnish content within the mixture the cohesive
force within the bed can be varied. For low vanish content fluidization is
expected, for high varnish content fracturing is expected. Thus a method of
exploring the transition between fracturing and fluidization is developed.
The cohesion can be found by doing shear tests as described earlier in
this thesis.
Famous last words
When you are as far as here in reading the thesis, I wish to say a few words
to you as a keen reader.
First of all I whish to thank you, as a keen reader, for paying me the
respect of reading my work. Secondly I will thank the people that have
helped me on the way. The discussions have been both fun and interesting.
May we have several discussions in the future! You know who you are, thank
you. :-)
“Alt har ein ende, så nær som pylsa. Ho heve to.”
Ivar Aasen
124
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131
Index
dissipative, 27
dolerite, 10
drag coefficient, 54
dykes, 9
dynamic viscosity, 61
Aarnes, I., 10
adiabatic compression, 79
advective derivative, 38
air supply, 62
angle of repose, 24, 25
arching, 27
aureole, 8, 10, 11
aureoles, 10
averaging problems, 33
earth, 22
effective capillary length, 51
effective permeability, 104
empirical classification, 33
Eocene, 15
Euler equation, 38
Euler’s equation, 43
Euler, L., 39
Eulerian derivative, 38
body force, 40
brecciation, 8, 9
bubbling, 99, 101, 108
capillary model, 50
capillary number, 46
Carman-Kozeny permeability, 51
cohesion, 25, 28, 60, 100, 112, 114
compressibility, 61
continuity equation, 39, 49
continum equations, 3
convection, 35
face center cubic lattice, 23
failure envelope, 29, 30
fluidization, 4, 7, 8, 13, 34, 53, 67,
68, 72–74, 76, 102, 108,
111–113
fluidizing, 3
force network, 27
force networks, 24, 33, 103
force probability distribution, 27
fracturing, 87, 88
fracturing vs fluidization, 95
friction, 27
frictional coefficient, 29
Darcy velocity, 49
Darcy’s law, 7, 17, 38, 47, 49, 64
Darcy, H, 47
Darcy, Henry, 64, 65, 73, 99, 102,
105, 115
deformation, 88
density, 60, 61
density-density correlations, 24
dilatancy, 22, 115
dimensional analysis, 73
dislocations, 32
granular liquids, 33
granular media, 3, 7, 21
granular solids, 22
Hagen-Poiseuille equation, 50
132
INDEX
planets, 21
plate tectonics, 22
porosity, 23, 51, 61, 73, 101, 104,
114
Hele-Shaw cell, 4, 59
hexagonal close packing, 23
hydrodynamic, 38
hydrofracturing, 8
hydrothermal vent complex, 7–9
random close packing (RCP), 23
random loose packing (RLP), 23
repulsive forces, 25
Reynolds number, 37, 38, 43,
99–101
Reynolds, O., 43
ideal fluids, 39, 40
isothermal compression, 79
Janssen wall effect, 22, 27, 28, 67
kimberlite, 4, 7, 12
kinematic viscosity, 43
Kozeny constant, 51
sedimentary basins, 8
segregation, 34, 93
specific surface area, 51
steady flow, 38
Stokes equation, 52
Stokes flow, 44, 45, 55
Stokes flow of a sedimenting
particle, 38
Stokes, G. G., 37, 50, 100
substantial derivative, 38
surface tension, 25
Lagrangian derivative, 38
Lagrangian volume, 38
maar craters, 8
magnetic field, 26
Mattson, Berit, 15
Mesozoic era, 11
mode 1 fracture, 32, 87, 100
mode 2 fracture, 31
Mohr circle, 29
Mohr diagram, 29
tensile fractur, 32
tensile fractures, 16
thermal fluctuations, 21
turtosity, 52
Navier, C.-L., 37
Navier-Stokes equation, 33, 37
Newton, 38, 102
Newtonian fluid, 41
nummulites, 11
venting in natural systems, 76
venting number, 13
viscosity tensor, 41
viscous drag, 8, 59, 68, 76, 102,
108, 112, 113
viscous force, 40
von Mises failure criterion, 32
Omega, 60
packing density, 23
palaeontology, 10
Paleocene, 10
permeability, 51, 65, 74, 76, 77
PGP, 4
phase diagram, 4
phenomenological constants, 41
pipelike structures, 9
Youngs modulus, 61
133
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