Magma Transport In Dikes - Rice University
Author's personal copyChapter 10Magma Transport in DikesHelge GonnermannDepartment of Earth Science, Rice University, Houston, Texas, USABenoit TaisneEarth Observatory of Singapore, Nanyang Technological University, SingaporeChapter Outline1. Introduction2. Overview3. Deformational Behavior of Materials3.1. Elastic Deformation3.2. Viscous or Plastic Deformation3.3. Viscoelastic Deformation3.4. Brittle Fracture4. Linear Elastic Fracture Mechanics4.1. Elastic Pressure4.2. Fracture tle A material that has little capacity to deform plastically and isnot able to relax stresses that become localized at crack-like defects during an applied deformation. In such a material, suchdefects will propagate rapidly and spatially localized deformationthat is brittle fracture. Such brittle deformation involves thebreaking of chemical bonds, whereas plastic and viscous deformation involves microscopic motion of atoms or molecules pastone another.buoyancy Buoyancy is a force induced by the density differencebetween two bodies. In the case of dikes, the buoyancy force isproportional to the density of the host rock minus the density ofthe magma. Therefore, this force can be either positive, and drivethe magma toward the surface, or negative, and prevent the ascentof the magma. The level at which the buoyancy changes sign iscalled the level of neutral buoyancy.buoyancy length The length of a dike at which its buoyancy force issufficient to result in its upward propagation without the need ofadditional magma supply from a pressurized reservoir.constitutive equation A material-specific relation between twophysical quantities, such as the response of a material to anexternally applied force.5. Magma Flow in Dikes6. Dike Formation and Propagation6.1. Dikes within Melting Zones6.2. Buoyancy-Driven Dike of Finite Volume6.3. Dike Connected to a Magma Source7. The Fate of Dikes7.1. The Level of Neutral Buoyancy7.2. Ambient Stress Field7.3. Thermal and Viscous EffectsFurther Reading219219219219220221221222222224crack A crack is a defect around which atomic bonding is completelybroken. This results in a large volume expansion and stress concentration and, hence, a long-range stress field surroundingcracks.cubic law The volumetric flow rate of a Newtonian viscous fluidthrough a planar channel of high aspect ratio and that is boundedon both sides by solid walls and scales with the cube of channelthickness. The functional relationship between flow rate andviscosity, pressure gradient, and channel thickness cubed is calledthe cubic law.elastic Upon an applied stress a material will deform. If the deformation is instantaneous and recoverable it is called elastic.feeder dike Dikes that supply magma to the surface during volcaniceruptions.fracture Interaction among cracks during an applied stress leads tolocalized deformation that is discernible macroscopically, that isbrittle fracture. In other words, brittle fracture starts with theformation of microcracks, which interact to form a macroscopically discernible fracture.fracture toughness The fracture toughness, specific to the materialand opening mode (see mode I definition), is the capacity of thematerial to resist fracturing when stressed.The Encyclopedia of Volcanoes. -9Copyright ! 2015 Elsevier Inc. All rights reserved.The Encyclopedia of Volcanoes, Second Edition, 2015, 215e224215
Author's personal copy216PART IGriffith’s theory According to Griffith’s theory, an existing crackwill propagate if the strain energy released is greater than thesurface energy created by the formation of the two new materialsurfaces.Hooke’s law The constitutive equation for a material that undergoesa recoverable deformation where the strain is linearly proportionalto the applied stress.level of neutral buoyancy (LNB) Theoretically, the level at whichbuoyantly ascending magma has the same density as the surrounding rock.linear elastic fracture mechanics (LEFM) Fracture mechanics isthe study of the propagation of cracks in elastic materials wherethe relationship between applied stress and strain is linear. Thebasic assumptions of LEFM are (1) the presence of crack-likedefects; (2) a crack is a free, internal, planar surface withstresses near the crack tip that can be calculated from linearelasticity; and (3) the growth of such cracks leads to the failure,that is, brittle fracture, of the material.lithosphere The lithosphere can be defined based on seismic,mechanical, or thermal considerations. The seismic lithosphere ofthe Earth was initially defined as a static unit lying above a lowvelocity zone for shear waves in the upper mantle, presumablyassociated with partial melting and weak mechanical behavior,that is, the asthenosphere. The mechanical lithosphere comprisesthose rocks that remain a coherent part of the plates on geologicaltimescales and deform in an elastic manner, even on long timescales, whereas the underlying asthenosphere undergoes permanent (plastic) deformation. Studies of flexure and deformation dueto topographic loads, such as volcanic edifices and mountainbelts, have led to a definition of lithosphere as the layer thatsustains elastic stresses to support these loads. The thermal lithosphere is taken to be the upper thermal boundary layer of aplanet, where heat is transported by conduction, as opposed to theunderlying asthenosphere where convection is the dominant heattransfer mechanism. Because the mechanical properties of rocksare strongly temperature-dependent, the mechanically and thermally defined lithospheres are essentially equivalent.Maxwell model A model for the deformational behavior of viscoelastic materials, which is based on the timescale at whichdeformation changes from recoverable to nonrecoverable.mode I There are three ways of applying a force to enable crackpropagation. One is a tensile stress normal to the plane of thecrack and the two others are shear stresses parallel to the plane ofthe crack. The mode in which a crack opens under tensile stressnormal to the plane of the crack is called mode I, whereas theother two are called mode II and mode III.Newtonian viscosity If for a viscous material the relationship between applied shear stress and resultant shear rate is linear, theconstant of proportionality is called the Newtonian viscosity.plastic If a solid material undergoes a time-dependent and nonrecoverable deformation under an applied stress it is calledplastic. The deformation may be distributed homogeneouslythroughout the material or it may be localized in a narrow region.stress intensity factor The growth of a crack under an applied stressis related to the tensile stress acting at the crack tip. For a givenopening mode, the relationship between applied stress and cracklength is characterized by a constant, called the stress intensityfactor.Origin and Transport of Magmathermal boundary layer It corresponds to the layer of magmaclosest to the dike walls where heat transfer is dominated byconduction toward the dike wall, as opposed to along-dikeadvection by magma flow.viscoelastic A material that undergoes recoverable deformation onshort timescales and nonrecoverable deformation on long timescales is called viscoelastic. The time at which deformationchanges from recoverable to nonrecoverable is often referred to asthe Maxwell time.viscous Under an applied stress a viscous fluid undergoes a timedependent, nonrecoverable deformation that is homogeneousthroughout the fluid volume.yield stress Upon an applied stress a material will deform. If thematerial will undergo recoverable deformation below somethreshold stress value and permanent deformation above thisvalue, this threshold stress value is called the yield stress. Afteryielding, nonrecoverable deformation is usually referred to asplastic deformation. However, sometimes the terms viscous andplastic deformation are used interchangeably.1. INTRODUCTIONDikes are the principal mode of magma transport withinand through planetary lithospheres. Dikes representthe critical link between regions of melt production andregions of melt accumulation, such as magma chambers.In some cases, dikes propagate to a planet’s surface,resulting in fissure eruptions. In other cases, dikes maygive way to more spatially localized conduits at shallowdepths, supplying magma to individual volcanic vents.This chapter provides a summary of the mechanics andFIGURE 10.1 Aerial view of a 8-km-long dike that radiates from ShipRock, New Mexico, USA, a volcanic plug that rises 550 m above the valleyfloor. Three prominent dikes radiate out from Ship Rock (Nature/UniversalImages Group/Getty Images).The Encyclopedia of Volcanoes, Second Edition, 2015, 215e224
Author's personal copyChapter 10Magma Transport in Dikesdynamics of dike propagation and magma transportthrough dikes.2. OVERVIEWDikes and sills are sheet-like bodies of igneous rock thatwere emplaced as magma within preexisting rock. They aretypically of the order of 1e10 m in thickness and of theorder of 1e100 km in lateral extent (Figure 10.1). Dikes areformed by fracturing the host rock, due to magma injection(Figure 10.2). Their propagation is oriented preferentiallyin the direction perpendicular to the minimum totalcompressive stress of the surrounding rock, but may alsofollow preexisting planes of weakness, such as beddingplanes or preexisting fractures.The formation and propagation of dikes may allow therapid transport of magma without extensive solidificationdue to cooling. Estimates of dike propagation speeds,deduced from seismic observations, are of the order ofmeters per second. In order for dikes to propagate, magmaneeds to exert sufficient stress at the dike tip to overcomethe fracture strength of the rock. Furthermore, magmapressure within the dike needs to exceed the stressesexerted by the surrounding rock on the dike walls, in orderto keep the dike from closing. Pressure gradients within themagma also need to balance viscous resistance, in order toallow magma to flow into the dike as it grows in length.Processes that may result in the required magma pressuresmay include the change in volume associated with melting,compaction of partially molten rock, magma buoyancyrelative to the surrounding rock, and in the case of dikesemanating from inflated magma reservoirs, the elastic energy stored in the reservoir’s wall rock, commonly knownas reservoir overpressure.Theoretical treatments of dike propagation thereforehave to consider the fracture mechanics at the dike tip,magma flow within the dike, deformation of the surrounding wall rock, as well as cooling of the dike (Rubin,1995). We will define several characteristic pressure scales,in order to facilitate the distinction of dominant stressbalances for dike propagation. End-member cases forpropagating dikes are (1) dikes of finite volume that are“self-propagating” due to magma buoyancy; (2) dikes thatpropagate within partially molten rock and are suppliedwith melt from the surrounding porous wall rock; and (3)dikes that are connected to a magma reservoir and aresupplied by magma from within the reservoir.3. DEFORMATIONAL BEHAVIOROF MATERIALSUpon an applied stress a continuous material will deform.For a fluid of Newtonian viscosity, the resultant217deformation is nonrecoverable. For a linearly elastic solidthe deformation is instantaneous and recoverable. If thesolid deforms elastically below some threshold stress, butyields to nonrecoverable deformation above this yieldstress it is called plastic. Nonrecoverable deformation canalso be brittle and localized, which involves the breakingof chemical bonds and is called fracture.3.1. Elastic DeformationThe constitutive equation for linear elastic deformation iss ¼ Me;(10.1)where s is the applied stress, M is the elastic constant, and eis strain. This relation is also called Hooke’s law. There arefive elastic constants. For example, Young’s modulus, E,applies to the case of uniaxial stresss ¼ Ee(10.2)and the shear modulus, m, in the case of shear stresss ¼ me:(10.3)The relationship between E and m is given byE ¼ 2mð1 þ nÞ;(10.4)where n is the Poisson’s ratio.3.2. Viscous or Plastic DeformationThe constitutive equation for linear viscous deformation iss ¼ h e;(10.5)where h is the Newtonian viscosity and e is the strain rate.For many natural materials, the relation of stress to strainrate is not linear and constitutive models for such nonNewtonian rheology often relate stress to strain rate via apower law.3.3. Viscoelastic DeformationIn addition many materials behave like an elastic solid onshort timescales, but viscously over longer times. The mostwell-known constitutive model for such viscoelasticdeformation is the Maxwell model, where the total strainrate is the sum of an elastic strain rate and a viscous strainrate, with the latter becoming increasingly dominant attimes greater than or equal to the Maxwell time.3.4. Brittle FractureUnder application of a stress, the fragmentation of a solidvolume into two or more parts is referred to as fracture.Although plastically deformable solids can undergoThe Encyclopedia of Volcanoes, Second Edition, 2015, 215e224
Author's personal copy218PART IOrigin and Transport of Magmaz aHost rockzyx2az 02bOffset bed or veinDilationDilationP P Dikez –aFIGURE 10.2 Schematic illustration of a dike with wall displacementsthat correspond to dilation by magma injection, as indicated by the offsetof preexisting planar features. This type of fracture wall displacement iscalled opening mode or mode I displacement.ductile fracture, we will focus, herein, on brittle fracture.The brittle fracture of a solid body typically involves theaccumulation of damage, resulting in the nucleation ofcracks or voids and finally, their growth. On a macroscopic level, the mechanical properties of solids tend tobe limited by their imperfections, also known as defects.Linear elastic fracture mechanics (LEFM) provides aconceptual framework that is based on crack propagationfrom such defects, in order to asses a material’s fracturetoughness. Because the in situ stresses at depth are nevertensile, dike formation and propagation requiresthat magma pressure exceeds the least principal stressby some critical amount in order for a tensile fractureto grow.4. LINEAR ELASTIC FRACTUREMECHANICSLEFM provides a framework for quantifying the stressrequired to propagate a fracture, through the use of aparameter known as the fracture toughness.4.1. Elastic PressureConsider a two-dimensional crack (Figure 10.3) of length2a and subject to a uniform excess internal pressure Ps,relative to a uniform remote stress s. This crack will have ahalf thickness ofFIGURE 10.3 Geometry of an elastically walled dike of infiniteextent in the y direction. The dike is of height 2a, width 2b, and has aninternal pressure of P that exceeds the stress in the surrounding rock iss by Ps. !"ð1 % nÞPs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 % z2 z a;b z ¼m(10.6)where z is the vertical coordinate. From the pressurerequired to open such an idealized crack by 2b at z¼0, onecan define a characteristic elastic pressure asPe wmb:ð1 % nÞ a(10.7)4.2. Fracture PressureAccording to Griffith, an existing crack will propagate ifthe strain energy released is greater than the surface energy created by the formation of the two new crack surfaces. The practical application of Griffith’s theory isdifficult, in part because of the challenges posed bymeasuring quantities such as surface energy. Instead material fracture is conventionally treated within the framework of LEFM, which postulates the existence ofmicrocracks within any real solid volume. Because stressbecomes concentrated at the tips of these microcracks theycan grow within a process zone located at the fracture tip,resulting in the opening and propagation of a macroscopicfracture.As a consequence of these material imperfections,actual fracture strengths are considerably lessdby aboutone to two orders of magnitudedthan the theoreticalcohesive material strength, also called cleavage stress.Moreover, the fracture of a material can be predicted interms of the tensile stress acting in the crack tip region.The Encyclopedia of Volcanoes, Second Edition, 2015, 215e224
Author's personal copyChapter 10Magma Transport in Dikes219Under pure tensile failure (mode I) this stress is proportional to the stress intensity factor, defined aspffiffiffiffiffiffiK ¼ YDPs pa;(10.8)where Y is a constant that depends on the crack openingmode and geometry, and for the geometry under consideration has a value of 1. The minimum value of K at whichfracture occurs or a crack propagates can be determinedexperimentally and is known as the fracture toughness, Kc.Laboratory measurements at atmospheric conditions indicate that Kcw1 MPa m1/2 and that it increases with pressure, although the actual fracture toughness under fieldconditions remains poorly resolved and the range of plausible values is from about 1 MPa m1/2 to 100 MPa m1/2. Thecharacteristic fracture pressure scale is thus defined asKcPf wpffiffiffi :a(10.9)5. MAGMA FLOW IN DIKESMost treatments of magma flow within dikes approximatethe flow as one-dimensional, laminar, and incompressible.In the case of a vertically oriented dike, the volumetric flowrate isQ ¼ 2bwuz ¼ %w 3 dP0b;3h dz(10.10)where uz is the average magma velocity in the z-direction,0h is the magma viscosity, P the nonhydrostatic magmapressure, and w is the dike breadth, which is the dimensionperpendicular to the direction of propagation (z) and to thewidth (x) of the dike. Equation (10.10) yields the characteristic viscous pressure scalePh whaQ hauw:b3 w b 2(10.11)Furthermore, using the average magma velocity, uz ,together with mass balance results in an equation for thechange in dike widthdb d ! "þbuz ¼ 0:dt dz(10.12)6. DIKE FORMATION ANDPROPAGATION6.1. Dikes within Melting ZonesThe isotopic disequilibria found in mid-ocean ridge, OceanIsland and island arc basalts require average melt ascentrates of 1e50 m yr%1 (Kelemen et al., 1997; Turner andBourdon, 2011). It is thought that these high ascent ratesand isotopic disequilibria are most likely a consequence ofchannelized porous flow by reactive melt transport withinthe asthenosphere. However, there are other processes thathave the potential to enhance rates of melt transport inmelting zones, such as the formation of melt-filled veinsduring deformation of the partially molten rock (Kohlstedtand Holtzman, 2009).To what extent dikes contribute to melt transportwithin melting zones, however, remains unclear (Kelemenet al., 1997; Weinberg, 1999). In the case of highlyviscous felsic melts, the flow of melt into the dike fromthe surrounding partially molten rock is very slow and ithas been suggested that a viable alternative could be thatupward heat transfer may facilitate pervasive meltmigration through ductile deformation, rather than elasticfracture (Weinberg, 1999).Dike growth in partially molten peridotite could be aconsequence of excess pore pressure due to the volumeincrease associated with melting and compaction(McKenzie, 1984). Melt could thus be forced into preexisting melt pockets and exceed the rock fracture toughness,thus facilitating fracture propagation (Rubin, 1998). If dikegrowth in partially molten peridotite is treated as a purelyelastic process, it is found that the upward propagation andwidth of dikes are primarily controlled by the balancebetween the rate of porous inflow from the surroundingrock and upward flow of melt within the dike. Because themelt flux into the dike is, to first order, dependent on theratio of permeability to melt viscosity the porous influx forfelsic dikes is rate limiting, making dike propagationdifficult. For asthenospheric dikes the balance betweenporous inflow and upward propagation results in theoreticaldike widths of the order of 10%2 m (Rubin, 1998). Suchdikes, if they exist, would be too thin to propagate withinthe lithosphere
3.1. Elastic Deformation 217 3.2. Viscous or Plastic Deformation 217 3.3. Viscoelastic Deformation 217 3.4. Brittle Fracture 217 4. Linear Elastic Fracture Mechanics 218 4.1. Elastic Pressure 218 4.2. Fracture Pressure 218 5. Magma Flow in Dikes 219 6. Dike Formation and Propagation 219 6.1. Dikes within Melting Zones 219 6.2. Buoyancy-Driven .
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Earth Science 11e Tarbuck/Lutgens. Earth Science, 11e Volcanoes and Other Igneous Activity Chapter 9. Volcanic eruptions Factors that determine the violence of an eruption Composition of the magma Temperature of the magma Dissolved gases in the magma Viscosity of magma
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