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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical background 7 9 12 13 16 19 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 22 23 24 24 27 27 29 30 30 32 32 33 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment 37 38 40 43 43 44 46 47 49 50 50 51 53 54 57 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 59 60 62 63 64 64 65 67 69 73 73 76 79 . . . . . . . 83 83 85 85 86 88 89 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 . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 93 94 95 . . . . . . . . . . 99 99 100 100 102 104 107 108 110 112 113 117 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. 81 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 Bibliography [1] S. McNamara and W. R. Young. Dynamics of freely evolving, two-dimensional granular medium. Physics of Fluids A, 4(496), 1992. [2] L. Bocquet, W. Losert, D. Schalk, T. C. Lubensky, and J. P. Gollub. Granular shear flow dynamics and forces: Experiment and continum theory. Physical Review Letters E, 65, 2002. [3] T. S. Woolsey, M. E. McCallum, and S. A. Schumm. Modeling of diatreme emplacement by fluidization. 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Journal of Volcanology and Geothermal Research, 2006. 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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