Laboratory Modelling of Volcano Plumbing Systems: A...

Laboratory Modelling of VolcanoPlumbing Systems: A Review

Olivier Galland, Eoghan Holohan,Benjamin van Wyk de Vries and Steffi Burchardt

AbstractWe review the numerous experimental studies dedicated to unravelling thephysics and dynamics of various parts of a volcanic plumbing system.Section 1 lists the model materials commonly used for model magmas ormodel rocks. We describe these materials’ mechanical properties anddiscuss their suitability for modelling sub-volcanic processes. Section 2examines the fundamental concepts of dimensional analysis and similarityin laboratory modelling. We provide a step-by-step explanation of how toapply dimensional analysis to laboratory models in order to identifyfundamental physical laws that govern the modelled processes indimensionless (i.e. scale independent) form. Section 3 summarises anddiscusses the past applications of laboratory models to understandnumerous features of volcanic plumbing systems. These include: dykes,cone sheets, sills, laccoliths, caldera-related structures, ground deforma-tion, magma/fault interactions, and explosive vents. We outline howlaboratory models have yielded insights into the main geometric andmechanical controls on the development of each part of the volcanic

O. Galland (&)Physics of Geological Processes (PGP), Departmentof Geosciences, University of Oslo, Oslo, Norwaye-mail: [email protected]

E. HolohanGerman Research Centre for Geosciences (GFZ),Helmholtz Zentrum Potsdam, Potsdam, Germany

B. van Wyk de VriesObservatoire de Physique du Globe deClermont-Ferrand, Blaise Pascal University,Clermont-Ferrand, France

S. BurchardtDepartment of Earth Sciences, Center forExperimental Mineralogy, Petrology andGeochemistry, University of Uppsala, Uppsala,Sweden

Advs in VolcanologyDOI: 10.1007/11157_2015_9© Springer International Publishing Switzerland 2015

1

plumbing system. We conclude with some perspectives on the limitationsof past and current analogue modelling approaches, and on challenges tobe addressed by future research.

1 Introduction

Volcanic plumbing systems set the stage forvolcanic eruptions, by controlling the flow ofmagma into the vent. The term ‘volcanoplumbing system’ is here broadly defined as thestructural framework of pathways and storageregions through which magma travels on itsjourney from its source region to the Earth’ssurface. As the metaphor of a plumbing systemsuggests, the focus here is primarily on thetransport and storage of magma within relativelysolid or brittle Earth materials that characterisethe upper part of the Earth’s lithosphere.

The huge scale ranges that characterise vol-cano plumbing systems represent a challenge forunravelling the complexity of underlying pro-cesses. These processes act at length scales ofmicrons to thousands of kilometres, and over time

scales of milliseconds to millions of years. Ourobservational range is limited, e.g. to the humanlife (and working) time and the geological pro-cesses are often hidden under ground (Fig. 1).Earth scientists try to overcome these obstacles bystudying geological systems as observed in thefield. However, field examples are commonly thecompound result of a series of past and/oron-going geological events and processes. Thisaggregation of effects can make it difficult to graspclearly the roles of individual physical processesin the geological system’s evolution as a whole.

One way to tackle these obstacles is to repli-cate geological processes in controlled laboratoryexperiments. Hall (1815) conducted one of thefirst such experiments to provide a qualitativephysical explanation (horizontal shortening) forfolded rock strata observed in Scotland. Bydesigning experiments to study the formation of

Fig. 1 Schematic drawingillustrating the maincharacteristics of volcanoplumbing systems. Theseinclude dykes (Sect. 4.1),cone sheets (Sect. 4.2), sills(Sect. 4.3), laccoliths(Sect. 4.4), caldera-relatedstructures and intrusions(Sect. 4.5), magma-faultinteractions (Sect. 4.7), andexplosive volcanic vents(Sect. 4.8). See text fordetails

2 O. Galland et al.

diatremes, Daubrée (1891) first simulated pro-cesses underlying the development of volcanoplumbing systems. Since then, laboratory modelshave been a key tool for scientists seeking tounravel the physical processes that control thedevelopment and behaviour of volcanic andmagmatic systems (Fig. 2).

In this chapter, we provide a review of howlaboratory models, also called analogue models,have been designed and adapted to unravel thedynamics of shallow magmatic systems. We alsodescribe and discuss the application of the labo-ratory results to various features and stages inthese systems’ evolution. As with other disci-plines in Earth Sciences, laboratory modellinghas evolved to become increasingly quantitativein recent years. As we highlight below, this hasdrastically increased the capacity of analoguemodels to aid the interpretation of not onlyintrusive and structural relationships in ‘fossi-lised’ field examples, but also geodetic andgeophysical data from active volcanoes.

2 Model Materials

An important technical aspect of laboratorymodels of volcanic plumbing systems is a rele-vant choice of model materials. Before startingan experimental project, it is very important toknow exactly the mechanical properties of themodel materials to be used. This step is crucialfor: (i) designing the experimental apparatus,(ii) setting the initial and boundary conditions ofthe models, and (iii) quantitatively interpretingthe model results. A wide range of materials withknown mechanical properties is available fromthe literature, both for model magmas and modelhost rocks (Figs. 3 and 4; Tables 1, 2, 3 and 4).

2.1 Model Magma

In nature, magma viscosity depends on manyfactors, such as chemical composition, tempera-ture, dissolved volatile content (e.g. water), andtotal crystal content. Magma viscosity hence has awide range, within which a lowest viscosity end

member would be a high temperature,crystal-poor, ‘wet’ mafic magma (η * 10–100 Pa s) (Dingwell et al. 1993), and a highestviscosity end member be a low-temperature,crystal-rich, ‘dry’ granite magma (η up to 1017−18

Pa s) (Scaillet et al. 1997). Such wide range meansthat different materials are required to simulatemagmas of high or low viscosity (Table 2).

2.1.1 “High” Viscosity MagmaThe most commonly used material for simulatinghigh viscosity magma is a silicone polymer orputty called Polydimethylsiloxane (PDMS;Figs. 3 and 4a). The advantage of PDMS is that itis a Newtonian fluid at typical laboratory strainrates and temperatures (Weijermars 1986; tenGrotenhuis et al. 2002; Boutelier et al. 2008).This means that the relationship of the appliedshear stress to resultant shear strain rate follows apower law of exponent of 1 (i.e. is linea). Itsviscosity is hence a constant at a given temper-ature and has been measured at *1–3 × 104 Pa sat room temperature (Table 2; e.g. Weijermars1986; Corti et al. 2005). As temperature varies,the viscosity of PDMS follows an Arrhenius law(Hailemariam and Mulugeta 1998; Cagnard et al.2006), i.e., it decreases exponentially withincreasing temperature. The viscosity of PDMScan be easily measured by using a cylindricalviscometer, as described by e.g. Cobbold andJackson (1992) and Reber et al. (2013), or acapillary (extrusion) viscometer, as described bye.g. Hailemariam and Mulugeta (1998).

According to experimental requirements, theviscosity of PDMS can also be increased ordecreased by the addition of other materials.Increasing PDMS viscosity is achieved by mix-ing it with different proportions of small mineralparticles (inert fillers). The resulting mixture is asuspension of particles within a viscous matrix,the viscosity of which follows theEinstein-Roscoe law (Einstein 1906; Weijermars1986). Depending on the specific gravity of theparticles used, the bulk density of such a mixturecan be increased or decreased with respect topure PDMS. For example, addition of BaSO4increases the bulk density (ten Grotenhuis et al.2002), whereas addition of fine hollow glass

Laboratory Modelling of Volcano Plumbing Systems … 3

Fig. 2 a Drawing of the pioneering experimental appa-ratus designed by Hans Ramberg, at the University ofUppsala, Sweden. 1 model in centrifuge cup; 2 strobo-scopic light reflector; 3 TV camera; 4 TV receiver;5 stroboscope; 6 temperature and speed control cabinet;7 motor; 8 refrigerator unit. b Sections through Ramberg’scentrifuged model of silicone putty with powdered-waxlayer above, embedded in painter’s putty with sheets ofmodelling clay. Run 10 min at 1300 g, the experimentwas intended to model diapir rise. Nevertheless, the

similarity between the model outcome and igneouslaccoliths suggests that such models can be applied tothese igneous features. c Ramberg’s model of magma risedue to buoyant forces. Left Initial arrangement of model.Centre and right Cross-sections through vent of a KMnO4solution through overburden in model shown in left. Thevent runs vertically from one edge of the collapsedoriginal “magma” chamber. Run less than 1 min at some1000 g. After Ramberg (1967)

4 O. Galland et al.

beads (Boutelier et al. 2008) decreases it. Forlarge contents of inert fillers, however, the mix-ture becomes non-Newtonian and its stress tostrain relationship may instead follow a powerlaw with an exponent greater than 1. At verylarge contents of fillers, the mixture may becomevisco-elastic (i.e. display substantial elastic orrecoverable strains). Decreasing the viscosity ofPDMS is achieved by mixing it with oleic acid,such as described by e.g. Corti et al. (2005) andReber et al. (2013), although the PDMS-oleicacid mixture becomes sticky and challenging tohandle. These two procedures provide anenlarged analogue magma viscosity range ofbetween *7 × 103 and *5 × 105 Pa s, i.e.almost two orders of magnitude.

In addition, the commonly-used stiff PDMSputty is an end member of a suite of silicone oilproducts of variable, controlled viscosities(Takada 1990, 1994b; de Bremond d’Ars et al.2001; Watanabe et al. 2002). Those used bythese authors range between 0.8 × 10−3 and1.337 Pa s, but products with a still-broaderviscosity range are available on the market.Given their wide range of viscosities, siliconeoils can hence be used to simulate both highviscosity and low viscosity magmas without theneed for adding other materials (cf. Table 2).

2.1.2 “Low” Viscosity MagmaDuring the last two decades, several experimen-tal techniques have been developed to simulatelow-viscosity magmas. The simplest modelmagma is water; it has been mostly used forsimulating dyke and sill emplacement in gelatine(Tables 1 and 2; Hyndman and Alt 1987;McLeod and Tait 1999; Menand and Tait 2002;Walter and Troll 2003; Kavanagh et al. 2006;Menand 2008; Kervyn et al. 2009; Le Corvecet al. 2013). Water is an incompressible Newtonfluid of viscosity *10−3 Pa s. It is often dyed,such that an intruding dyke can be opticallytracked through models made of transparentgelatine. In these models, the density of theintruding water is very similar to that of thehosting gelatine. Consequently, an internal fluidpressure imposed by the experimentalist isrequired to trigger the propagation of the neu-trally buoyant intrusion.

To study buoyancy-driven fracturing, air isused as the intruding fluid (Table 2; Figs. 3 and4d–e). Air has a density of 1.2 × 10−3 kg m−3 anda viscosity of *2×10−6 Pa s at room temperature(e.g. Rivalta et al. 2005; Rivalta and Dahm2006). In addition to being much less dense thanthe host materials, air is compressible, such that arising experimental dyke is subject to pressuredecrease, and so to volume increase. In nature,magma compressibility (Rivalta 2010) and fluidexsolution (Menand and Tait 2001; Rivalta andDahm 2006; Taisne and Jaupart 2011) areexpected to lead to volume increase of propa-gating dykes, which is likely to affect theirpropagation dynamics. At the scale of theexperiments, however, Rivalta and Dahm (2006)estimated that this effect is negligible.

Air has also been used to study the formationof shallow magma conduits resulting fromexplosive processes due to rapid gas exsolutionof the rising magma (Haug et al. 2013; Gallandet al. 2014b), with applications to maar-diatremes(e.g. Woolsey et al. 1975) (see also Table 1;Figs. 3 and 4e), kimberlite pipes (e.g. Gernonet al. 2009), hydrothermal vent complexes(Nermoen et al. 2010a), and mud volcanoes (e.g.Mazzini et al. 2009).

Mag

ma

visc

osity

Rock rheology/strength

Gelatine/Air

Gelatine/Water

Gelatine/Silicone oil

Sand/Silicone

Ignimbrite/ Silica flour/

Silica flour/

Silica flour/Air

Golden Syrup

Vegetable oil

Sand/Air

Golden Syrup

Plastic/Viscous

Gas

/Bas

alts

Bas

alts

-Rhy

olite

sG

rani

tes

Elasto-plastic Elastic

Fig. 3 Qualitative diagram representing the mechanicalproperties of the main model materials that have beenused in laboratory models of volcano plumbing systems(see Tables 1, 2, 3 and 4). Note that for model rocks, theelastic, elasto-plastic and viscous/plastic rheologies cor-respond to the dominant behaviours of the correspondingmodel materials. See Sect. 2.2 for explanations


(a) (b)

(c) (d)

(e) (f)

6 O. Galland et al.

Model rocks in some models are made ofgranular media (see below and Table 4). In thesemodels, very low viscosity fluids like air andwater are unsuitable for the simulation of magmaintrusion, because they flow through the poresbetween the grains in addition to intruding themodel. Moreover, the most commonly usedgranular media are made of silica, which ishydrophilic, such that the percolation of waterthrough the pores is enhanced by surface tension.Therefore, other fluids such as oil (Grout 1945),honey, or Golden Syrup, have been used in thiscase (Table 2; Figs. 3 and 4b). The latter’s vis-cosity is 50 Pa s at room temperature, and itsdensity is 1400 kg m−3 (Mathieu et al. 2008;Kervyn et al. 2009; Abdelmalak et al. 2012).

Limiting the percolation of model magma inits host granular medium requires the use of afluid that is chemically incompatible with itshost, i.e. high surface tension. Good examplesare greases (Johnson and Pollard 1973; Pollardand Johnson 1973), which are hydrophobic, suchas the vegetable oil sold in France under thename Végétaline (Table 2; Figs. 3 and 4c)(Galland et al. 2006, 2009; Galland 2012). Thisfluid offers several advantages:

• its viscosity is low (η = 2 × 10−2 Pa s at 50 °C), such that it is suitable for simulating lowviscosity magmas (Galland et al. 2006);

• it is solid at room temperature, and its meltingtemperature is 31 °C. It hence can be injectedat molten state at a relatively low temperature(50 °C), that is technically easy to handle;

• it percolates very little into a veryfine-grained granular host medium;

• at the end of the experiments, it solidifies after15–30 min such that (i) the intrusions can beexcavated and their shapes analysed in 3D(Galland et al. 2009; Galland 2012), or (ii) themodels can be cut to reveal how intrusionsrelate to surrounding structures and overlyingsurface morphology (Galland et al. 2003,2007a, 2008).

The vegetable oil’s viscosity was measuredvia a rotary viscometer (Galland et al. 2006),which is a standard device to measure fluids ofrelatively low viscosities.

Finally, room temperature vulcanization(RTV) silicone (Gressier et al. 2010) has alsobeen used as a model magma (Table 2). Its vis-cosity is 25 Pa s. RTV silicone solidifies afterapproximately five hours, such that the simulatedintrusions can be excavated.

2.1.3 Conceptual MagmaChambers

Some experimental studies of volcanic plumbingsystems do not focus on the magma emplacementmechanisms, but on the deformation patternsinduced by inflating or deflating magma bodies(Table 1). In these cases, an inflating/deflatingmagma chamber is conceptually simulated with aballoon (Martí et al. 1994; Walter and Troll 2001;Troll et al. 2002; Geyer et al. 2006) or a movingpiston (Acocella et al. 2001; Burchardt andWalter 2010). Although these experiments do not

Fig. 4 Characteristic examples of laboratory models ofvolcano plumbing systems made by using various modelmaterials. a Schematic drawing of experimental setup (top)and photograph of model cross section (bottom) of thesand + silicone experiments of Román-Berdiel et al.(1995). b Side view photograph of ignimbrite + GoldenSyrup experiment (Mathieu et al. 2008). Ignimbritepowder simulates elasto-plastic rocks, whereas GoldenSyrup simulates magma of intermediate viscosity. c Pho-tograph and corresponding drawing of a cross-section ofsilica-flour + vegetable oil model (Galland et al. 2007a).The silica flour simulates the behaviour of the brittle crust,and exhibits faults, whereas the oil simulates magma oftypical viscosity. The cohesion of the silica flour allows theoil to form sheet intrusion. Note that the oil solidifies, such

that it becomes possible to make cross-sections through themodels. d Time series of side- and front-view photographsof gelatine + air experiments (Rivalta and Dahm 2006).The gelatine simulates elastic rock, whereas the airsimulates low viscosity magma. These experiments onlysimulate sheet intrusions (dyke and sills). e Time series ofside-view photographs of silica flour + air 2D experiments(Haug et al. 2013). The silica flour represents brittlecohesive rock, and the air represents rapidly degasingmagma or phreatomagmatically-generated water vapour.f Surface view photograph of model using conceptualmagma chamber (Troll et al. 2002). The inflation/deflationof a balloon represents the replenishment and draining of amagma chamber, and the deformation induced in theoverburden is observed at the surface of the models

b


Table 1 List of analog ‘rock’-‘magma’ combinations used in the experimental studies referenced in this chapter

Rock analogue Magmaanalogue

Part of plumbing system anddynamic regime modelled

References

Gels

Gelatin Air Ascent of low viscosity dykes(buoyancy driven)

Takada (1990), Bons et al. (2001),Menand and Tait (2001), Mulleret al. (2001), Ito and Martel (2002),Rivalta et al. (2005), Rivalta andDahm (2006), Le Corvec et al.(2013)

Hexane Interactions between twoascending dykes

Ito and Martel (2002)

Water Propagation or ascent ofintermediate viscosity dykes andsills (internal pressure driven)

Fiske and Jackson (1972),McGuire and Pullen (1989),Takada (1990), McLeod and Tait(1999), Menand and Tait (2001,2002), Walter and Troll (2003),Kavanagh et al. (2006), Menand(2008); Kervyn et al. (2009),Tibaldi et al. (2014), Daniels andMenand (2015)

Water-gelatinsolution

Dyke, sill and laccolithemplacement under volcanic cone

Hyndman and Alt (1987)

Mud Hydraulic fracturing in differentstress fields

Hubbert and Willis (1957)

Water-Natrosolsolution

Dyke nucleation from modelmagma reservoir

McLeod and Tait (1999)

Water-Sugarsolutions

Propagation and arrest ofbuoyancy-driven dyke

Taisne and Tait (2009), Taisneet al. (2011)

Water-Glycerin Dyke propagation and nucleation,and composite dyke emplacement

Koyaguchi and Takada (1994),McLeod and Tait (1999), Takada(1999)

Molten wax Cooling effects on dykepropagation

Taisne and Tait (2011)

Vegetable oil Cooling effects on dyke and sillemplacement

Chanceaux and Menand (2014)

Silicone oils Dyke propagation and nucleation Takada (1990, 1994b), McLeodand Tait 1999), Watanabe et al.(2002)

Grease Sill and laccolith emplacement Johnson and Pollard (1973),Pollard and Johnson 1973)

Silicone putty Dyke nucleation Cañón-Tapia and Merle (2006)

Acryl gel Air Dyke propagation driven bymagma pulses

Maaløe (1987)

Oil Dyke propagation Maaløe (1987)

Agar-sugar gel Water-glycerolsolution

Dyke propagation along level ofneutral buoyancy

Lister and Kerr (1991)

(continued)

8 O. Galland et al.

Table 1 (continued)



References

Granular materials

Sand or sandwith siliconelayers

Silicone putty Inflating or deflating laccoliths orstock-like intrusions, driven byinternal overpressures orunderpressures

Merle and Vendeville (1995),Román-Berdiel et al. (1995, 1997),Benn et al. (1998, 2000),Román-Berdiel (1999), Acocellaet al. (2000, 2001, 2004), Rocheet al. (2000, 2001), Girard and vanWyk de Vries (2005)

Rigid piston Deflating stock-like or sill-likeintrusions

Burchardt and Walter (2010)

Balloon Deflating sill-like or laccolithicintrusions

Walter and Troll (2001), Lavalléeet al. (2004), Geyer et al. (2006)

Silicone-Oleicacid mix

Magma emplacement in deformingcrust

Corti et al. (2005), Musumeci et al.(2005), Mazzarini et al. (2010),Montanari et al. (2010a, b), Ferréet al. (2012)

Glycerol Magma emplacement in tectonicextension

Bonini et al. (2001), Corti et al.(2001)

Air Explosive fragmentation of nearvent region

Walters et al. (2006)

Wheat flour Balloon Deflating and/or inflating (cyclic)sill-like or laccolithic intrusions

Walter and Troll (2001)

Glass beads Air Formation of hydrothermal ventcomplex

Nermoen et al. (2010a)

Air + glassbeads

Diatreme formation Ross et al. (2008a, b)

Crushedcohesive sand

Sleeve + plates Opening of dyke and associatedsurface deformation

Trippanera et al. (2014)

Sand-plastermix

Golden syrup Caldera collapse and dykepropagation

Roche et al. (2001), Kervyn et al.(2009), Delcamp et al. (2012b)

“Creamed”honey

Deflating sill-like intrusions Holohan et al. (2008a, b, 2013)

Ignimbritepowder

Golden Syrup Propagation and inflation of dykes,sills, and saucer-shaped intrusionsdriven by internal pressure

Mathieu et al. (2008), Mathieu andvan Wyk de Vries (2009)

Sand-flour mix Silicone putty Inflating intra-edifice stock-likeintrusions (cryptodomes), drivenby internal overpressures. Deflatingsill or stock-like magma chambers

Donnadieu and Merle (1998),Roche et al. (2000), Merle andDonnadieu (2000)

Balloon Deflating laccolithic magmachambers

Holohan et al. (2005)

Fused aluminapowder

Balloon Deflating and/or inflatinglaccolithic magma chambers

Martí et al. (1994)

Diatomitepowder

RTV silicone Dyke and sill propagation in thepresence of pore fluid overpressure

Gressier et al. (2010)

(continued)


simulate the complexity of volcanic plumbingsystems, the phenomenological results can bevery inspiring for understanding geologicalstructures observed in the field (Figs. 3 and 4f).

2.2 Model Rocks

Like magmas, rocks exhibit various mechanicalbehaviours depending on their lithology and onthe scales (both length and time) of the simulatedprocesses (Tables 3 and 4). Natural rocks exhibitcomplex rheologies, which combine elastic, vis-cous and plastic properties. Elastic deformation isreversible and usually occurs at lowstresses/strains. If the stress/strain exceeds therocks’ elastic limit, they deform in an irreversible

manner. They can do so by viscous behaviour,where the strain rate is a linear or power lawfunction of the applied stress, or by plasticbehaviour where the strain is a function of theapplied stress. Various rheological laws describ-ing these end member behaviours exist (Turcotteand Schubert 2002; Jaeger et al. 2009). In nature,rock rheology combines these different endmembers in a non-trivial manner, such that thecomplete rock behaviour is extremely challengingto characterise. Nevertheless, under given pres-sure and temperature conditions, length-scalesand time-scales (strain rates), rock behaviour isdominated by one or the other end member. Dif-ferent laboratory materials are available to simu-late these dominant rock-mechanical behaviours.

Table 1 (continued)



References

Silica powder Air Explosive fragmentation of nearvent region

Haug et al. (2013), Galland et al.(2014b)

Vegetable oil Propagation and inflation of dykes,sills, and saucer-shaped intrusionsdriven by internal pressure

Galland (2005, 2012), Gallandet al. (2003, 2006, 2007a, 2008,2009, 2011, 2014a), Ferré et al.(2012)

Golden Syrup Propagation and inflation of dykesdriven by internal pressure

Abdelmalak et al. (2012)

Sand-Powderedclay mix

Rigid body Deflating or inflating intrusions(idealised as sphere)

Komuro et al. (1984), Komuro(1987)

Balloon Deflating and/or inflating (cyclic)sill-like or laccolithic intrusions

Walter and Troll (2001), Troll et al.(2002)

Clay AqueousKMnO4

Buoyancy-driven dyke ascentthrough brittle crust

Ramberg (1970, 1981)

Others

Rigid walls Air + glassbeads

Mixing of materials in vents andconduits by gas-driven fluidization

Gernon et al. (2008, 2009)

Wet plaster AMS analysis on analogueintrusion

Závada et al. (2006, 2009, 2011)

Rigid piston Aqueouscorn-syrupsolutions

Ring dyke intrusion and internalflow synchronous with a deflatingsill-like intrusion

Kennedy et al. (2008)

Plasticine-wax Silicone putty Laccolith formation Dixon and Simpson (1987)

Wet clay Air Diatreme formation Grout (1945)

Water-cornsyrup mix

Air Diatreme formation Grout (1945)

10 O. Galland et al.

Table 2 Types and mechanical properties of analogue ‘magmas’ used in laboratory models of volcanic plumbingsystems

Model magma Macroscopic behaviour Viscosity Density References

Silicone polymers

PDMS(silicone putty)

Stiff Newtonian fluidat low strain rates(power law fluid at highstrain rates)Rigidity (shear)modulusG = 2.6 × 105 Pa

1–3 × 104 Pa s(at roomtemperature)

965–1140 kg m−3 Weijermars (1986),Acocella et al. (2001);ten Grotenhuis et al.(2002), Corti et al. (2005)

PDMS + inertfillers

Stiff Newtonian fluid iflow filler contentStiff non-Newtonianfluid if high fillercontent

3 × 104 to5 × 105 Pa s

Variable Weijermars (1986),Boutelier et al. (2008),Reber et al. (2013)

PDMS + oleicacid

Sticky Newtonian fluid 7 × 102 to1 × 104 Pa s

1060 kg m−3 Corti et al. (2003, 2005),Reber et al. (2013)

RTV silicone Viscous fluidSolidifies in contactwith air

25 Pa s 1500 kg m−3 Gressier et al. (2010)

Silicone oil Viscous NewtonianfluidIncompressible

Used:0.8 × 10−3 to1.337 Pa s (cancover broaderrange ofviscosities)

800–965 kg m−3 Takada (1990, 1994a, b),de Bremond d’Ars et al.(2001), Watanabe et al.(2002)

Water-sugar solutions

Corn-syrup Viscous Newtonianfluid

81 Pa s at 26 °C 1425 kg m−3 Kennedy et al. (2008)

“Creamed”honey

Intermediate viscosity,non-Newtonian(thixotropic) fluid

150–400 Pa s at20 °C,decreasing withincreased rotaryshear rate

1350 kg m−3 Holohan et al.(2008a, b)

Golden syrup Sticky viscous fluidIncompressibleNear Newtonian fluid

30–500 Pa s 1400 kg m−3 Mathieu et al. (2008),Kervyn et al. (2009),Beckett et al. (2011),Schellart (2011),Abdelmalak et al. (2012),Delcamp et al. (2012b)

Aqueouscorn-syrupsolutions

Low viscosityNewtonian fluid

0.0155–0.127 Pa s

1211–1305 kg m−3 Kennedy et al. (2008)

Glycerin–watersolutions

Low viscosityNewtonian fluid

10−3 to 10 Pa s 1000–1260 kg m−3 Takada (1990), Koyaguchiand Takada (1994), Boniniet al. (2001)

Concentratedsugar-watersolutions

Low viscosityNewtonian fluidIncompressible

– 1102–1373 kg m−3 Taisne and Tait (2009)

Water Low viscosityNewtonian fluidIncompressible

10−3 Pa s 1000 kg m−3 Hyndman and Alt (1987),Takada (1990), Menandand Tait (2002), Kavanaghet al. (2006)

(continued)


2.2.1 Plasticity-DominatedMaterials (Weak Materials)

The most commonly used material for simulatingmodel rocks is dry, loose, quartz sand (Tables 3and 4; Figs. 3 and 4). Its cohesion is negligible(Schellart 2000; Mourgues and Cobbold 2003),such that (i) it fails along shear zones (mode IIfractures), thereby simulating faults, and (ii) itdoes not allow open fractures (mode I fractures) toform. Failure of cohesion-less dry sand occursaccording to a Mohr-Coulomb criterion that is

defined, in theory, only by the angle of internalfriction (ϕ). Dry sand is most applicable for sim-ulating: (i) relatively low cohesion host materialsin nature, such as unconsolidated or weakly lith-ified sediments or tuffs, and/or (ii) large-scale(several to 100s of km) processes involvingregional-tectonic deformation (Corti et al. 2001,2003, 2005; Musumeci et al. 2005; Mazzariniet al. 2010; Montanari et al. 2010a, b), wherebythe material strength is assumed to decrease rela-tive to the increasing length-scale (Schultz 1996).

Table 2 (continued)

Model magma Macroscopic behaviour Viscosity Density References

Oils and waxes

Grease Viscous fluid – – Johnson and Pollard (1973)

Moltenvegetable oil

Low viscosityNewtonian fluidSolid at roomtemperatureIncompressible

2 × 10−2 Pa s at50 °C

890 kg m−3 Galland et al. (2006)

Molten wax Low viscosityNewtonian fluidSolid at roomtemperatureIncompressible

4 × 10−4 to7 × 10−2 Pa s

1100–1335 kg m−3 Taisne and Tait (2011)

Others

Mud Viscous fluid, nonNewtonian

– – Hubbert and Willis (1957)

Wet plaster Bingham pseudoplasticfluid

0.8–6 Pa s – Závada et al. (2009)

Water–Natrosolsolution

Low viscosity fluid 0.001–0.5 Pa s – McLeod and Tait (1999)

AqueousKMnO4solution

Low viscosity dyedNewtonian fluidIncompressible

10−3 Pa s – Ramberg (1970, 1981)

Aqueousmethyleneblue-sodiumpolytungstatesolution

Low viscosity dyedNewtonian fluid,Incompressible,adjustable density

10−3 Pa s 996–1010 kg m−3 Hallot et al. (1994, 1996)

Hexane Low viscosityNewtonian fluid(toxic for long-termexposure)

3 × 10−4 Pa s 1330 kg m−3 Ito and Martel (2002)

Air Low viscosity gasCompressibleLow density

2 × 10−6 Pa s 1.2 × 10−3 kg m−3 Takada (1990), Rivaltaet al. (2005)


Table

3Typ

esandmechanicalprop

ertiesof

non-granular

viscoelastic

andviscop

lastic

materialsused

asanalog

ue‘rock’

inlabo

ratory

mod

elsof

volcanic

plum

bing

system

s

Rock

analog

ueMacroscop

icbehaviou

rElastic

prop

erties

Fracture

toug

hness

Density

Viscosity

References

Gels

Gelatin

Viscoelastic

solid

Prop

ertiesdepend

on:

temperature,gelatin

concentration,

strain

rate,

andage(tim

eelapsed

aftersetting

)

E=39

0–16

,000

Paν=0.5

28–

200Pa

m1/2

1000–

1010

kgm

−3

Com

plex

viscosity

atconcentrationof

2.5

wt%

,temperature

of10

°C(gel-state),and

strain

rate

of10

−2s−

1isc.

50Pa

s.At30–

40°C

(sol-state),theviscosity

liesbetween

0.36–0.00

15Pa

s,decreasing

with

increased

strain

rate

intherang

e0.01–12

s−1

Djabo

urov

etal.

(198

8b),T

akada(199

0),

Rivalta

andDahm

(200

6),DiGiusepp

eet

al.(200

9),Kavanagh

etal.(201

3)

Acryl

amid

gel

Gel

ofvariable

rheology

from

nearly

liquidto

nearly

brittle

material

(poisono

us)

––

––

Maaløe(198

7)

Aqu

eous

sugar-agar

gels

Viscoelastic

solid

Prop

ertiesdepend

on:

temperature,g

elandsugar

concentrations,waitin

gtim

e

E=10

00Pa

20Pa

m1/2

1066–

1310

kgm

−3

−ListerandKerr(199

1)

Water

New

tonviscou

sfluid

−−

1000

kgm

−3

10−3Pa

sGrout

(194

5)

Others

Silicon

epu

ttyNew

tonviscou

sfluid

G=3×10

4

to2.6×10

5

Pa

−10

4Pa

sRam

berg

(198

1),

Rom

án-Berdiel

etal.

(199

5),Cortiet

al.

(200

3),Bou

telieret

al.

(200

8)

Plasticine

Power

law

material

(exp

onentn=6–9.5)

−−

1770

kgm

−3

–McC

lay(197

6),

Weijerm

ars(198

6),

Dixon

andSimpson

(198

7)

Wax

Power

law

material

(exp

onentn=2)

E=2×10

6

to10

7Pa

−95

0kg

m−3

−Dixon

andSimpson

(198

7)

Water-Corn

syrupmix

Viscous

fluid

Tem

perature-dependent

viscosity

−−

1211–

1305

kgm

−3

0.3–20

Pas

Grout

(194

5),Kennedy

etal.20

08)

Wet

clay

Visco-elasto-plastic

material

−−

−−

Grout

(194

5)


Note that although dry sand is very weak, itdoes not exhibit a purely or ideally plasticbehaviour. At low stresses and strains, it holdssome reversible, elastic deformation, as demon-strated by the measurements of, e.g. Lohrmannet al. (2003) and Panien et al. (2006). Moreover,under loading conditions typical of laboratory

experiments, dry sand fails in a semi-brittlefashion characterised by a stress drop that reflectsa difference in static and dynamic friction coef-ficients. Such properties can become importantin, e.g. models made of dry rice, which intendto simulate stick-slip faulting (Rosenau et al.2010).

Table 4 Types and mechanical properties of granular materials used as analogue ‘rock’ in laboratory models ofvolcanic plumbing systems

Rock analogue Macroscopicbehaviour

Cohesion(estimated)

Frictionangle

Density References

Sand LooseCoulombmaterial

12–123 Pa 25–35° 1300–1700 kg m−3 Schellart (2000),Lohrmann et al.(2003), Mourguesand Cobbold (2003)

Wheat flour SlightlycohesiveCoulombmaterial

40–50 Pa(likelyunderestimated)

33° 570 kg m−3 Walter and Troll(2001), Holohanet al. (2008a)

Sand-Plastermix

SlightlycohesiveCoulombmaterial

40–100 Pa(likelyunderestimated)

36° 1360–1560 kg m−3 Roche et al. (2001),Kervyn et al. (2009,2010)

Glass beads LooseCoulombmaterial

64–137 Pa 20° 1547–1905 kg m−3 Schellart (2000),Nermoen et al.(2010a, b)

Ignimbritepowder

CohesiveCoulombmaterial

100–230 Pa 38° 1400 kg m−3 Mathieu et al.(2008)

Sand-flour mix CohesiveCoulombmaterial

200 Pa 40° 1200–1400 kg m−3 Donnadieu andMerle (1998),Holohan et al.(2005)

Fused aluminapowder

CohesiveCoulombmaterial

200 Pa 38° – Martí et al. (1994)

Diatomitepowder

Low densitycohesiveCoulombmaterial

300 Pa – 400 kg m−3 Gressier et al.(2010)

Silica flour Cohesivefine-grainedCoulombmaterial

300–400 Pa 40° 1050–1700 kg m−3 Galland et al. (2006,2009), Abdelmalaket al. (2012)

Sand-powderedclay

CohesiveCoulombmaterialLikelyanisotropic

500–1300 Pa(likelyoverestimated)

26–31° 1500–1700 kg m−3 Komuro (1987),Walter and Troll(2001)


2.2.2 Elasticity-DominatedMaterials (Strong Materials)

While sand is a weak end member of modelrocks, the strong end member is gelatine(Table 3; Figs. 3 and 4d). At the scales of labo-ratory models, gelatine fails through the forma-tion of open cracks (mode I fractures). In thesemodels, gelatine behaves elastically, exceptlocally at the tips, where complex viscoelasticprocesses occur. Hence gelatine is suitable forsimulating: (i) natural host rocks of relativelyhigh cohesion, such as well-consolidated sedi-mentary rocks or crystalline igneous or meta-morphic basement rocks, and/or (ii) relativelysmall-scale processes (10s of m to a few km).Another advantage of gelatine, and gels in gen-eral, is that they become birefringent whenstrained (e.g. Dupré and Lagarde 1997; Dupréet al. 2010). When the models are placedbetween crossed polarizers, the strained domainsof the models appear as coloured fringes,whereas unstrained domains appear dark (Rich-ards and Mark 1966; Pollard and Johnson 1973;Taisne and Tait 2009, 2011).

The stiffness (Young’s modulus, E) and thestrength (fracture toughness, Kc) of gelatine canbe jointly tuned by varying the gelatine concen-tration (Di Giuseppe et al. 2009; Kavanagh et al.2013), while the Poisson’s ratio is constant atν = 0.5 (Djabourov et al. 1988a, b). Kc can becalculated by measuring the fluid pressurerequired to propagate an existing crack (Menandand Tait 2002; Kavanagh et al. 2013). Systematicmeasurements show that E is a linear function ofthe gelatine concentration (Kavanagh et al.2013). Nevertheless, for a given gelatine con-centration, E is also time-dependent (Kavanaghet al. 2013), as the gelification process continueslong time after the gelatine becomes solid. Thevalue of Kc is a function of

ffiffiffiffiE

p, and therefore of

the gelatine concentration also (Kavanagh et al.2013). This means that the preparation of gela-tine models must follow a rigorous procedure,during which the waiting time between thepreparation of the model and the experimentalrun is constant to ensure repeatable mechanicalproperties of the models.

2.2.3 Elasto-Plastic (IntermediateStrength) Materials

Dry sand and gelatine are the endmembers of rockmechanical behaviours, being plasticity- andelasticity-dominated, respectively. Nevertheless,it is known that complex processes, in whichelastic and plastic behaviours are balanced, cangovern natural rock deformation. Therefore, othermodel materials are needed to explore the effectsof such complex rock rheology on magmaemplacement in the Earth’s crust. Good examplesare cohesive granular materials (Table 4; Fig. 4b,c, e, f), such as ignimbrite powder (Mathieu et al.2008, 2011;Mathieu and vanWyk deVries 2011),fine-grained silica flour (Galland et al. 2006, 2009;Galland 2012), diatomite powder (Gressier et al.2010), or sand-plaster mixtures (Roche et al.2001), among others. When deformed, thesematerials typically exhibit both open (mode I) andshear (mode II) fractures at shallow and deeplevels, respectively (Holland et al. 2006; Ketter-mann and Urai, 2015). Although their elasticproperties remain challenging to estimate, thesegranular materials thus appear to be most suitablefor reproducing the complex elasto-plasticmechanical behaviour of natural rocks.

Such cohesive granular materials also failaccording to a Mohr-Coulomb criterion, but onethat is defined by the material’s tensile strength(T), cohesion (C) and its angle of internal friction(ϕ). These parameters control the depth of thetransition from Mode 1 to Mode 2 fracturing.The cohesion and the angle of internal frictionare properties that are measured indirectly.Firstly, a Mohr-Coulomb failure envelope isconstructed via numerous shear tests conductedby using either a Hubbert-type shear box (Hub-bert 1937; Mourgues and Cobbold 2003; Gallandet al. 2006) or a Casagrande shear box (Rossi andStorti 2003). These apparatuses measure theshear strength τ of the tested material for variablenormal stress σn. In a diagram displaying τ as afunction of σn, many measurements exhibit apositive correlation. By fitting a function or‘failure envelope’ to the data, it is possible tocalculate C as being the intercept of the fittingfunction with the shear stress axis (typically the


y-axis in such a diagram; e.g. Schellart 2000;Mourgues and Cobbold 2003). The slope of theenvelope, which is typically linear for most of itslength, is the tangent of Φ. The tensile strengthcan be measured directly (Schweiger and Zim-mermann 1999), but this requires specific devi-ces, such that tensile strengths are rarely reported(Galland et al. 2006; Holland et al. 2006).

The degree of compaction (or closeness ofpacking) of a granular material strongly influ-ences its mechanical properties (Lohrmann et al.2003; Galland et al. 2006; Schreurs et al. 2006).Increased compaction leads to an increaseddensity, cohesion and friction coefficient. Thedegree of compaction is controlled by the pro-cedure of emplacing the material in the experi-mental apparatus. Compaction can be increasedby shaking the model before running it, espe-cially if the material is cohesive (Galland et al.2009; Galland 2012), or by sifting the material ifit is cohesionless (Lohrmann et al. 2003; Maillot2013). Simply pouring the granular material inthe experimental box is to be avoided as it leadsto inhomogeneity in initial packing and hence inmechanical properties (Lohrmann et al. 2003).

2.2.4 Other MaterialsSome of the first experiments of magmaemplacement considered the whole crust as aviscous fluid. Grout (1945) injected air into mediaof various viscosities, such as wet clay, cornsyrups and water-diluted syrups in order to sim-ulate structures related to batholith emplacement.The rheologies of these pioneering model mate-rials were not accurately measured, however.Silicone putty (Fig. 4a) is commonly used tosimulate viscously deforming rock strata or for-mations, such as evaporite and shale(Román-Berdiel et al. 1995, 1997; Román-Berd-iel 1999; Corti et al. 2001, 2003). Nevertheless, itbecomes technically challenging to simulate bothviscous host rock and the injection of viscousmagma in the same experiments.

2.2.5 SummaryMost existing experimental studies of volcanicplumbing systems have used end member modelrock materials with rheologies that are

elasticity-dominated, plasticity-dominated, orviscous. Recently, the introduction of cohesivegranular materials has allowed for exploring theeffect of complex elasto-plastic behaviour of hostrock on magma transport and emplacement.Moreover, although highly concentrated gels areelasticity-dominated, low-concentration gels (e.g.laponite, gelatine, agarose) exhibit viscoelasticbehaviour at the length and time scales of labo-ratory experiments (e.g. Hallot et al. 1996; DiGiuseppe et al. 2009; Ruzicka and Zaccarelli2011) and offer the possibility to explore experi-mentally the emplacement mechanism of magmainto viscoelastic rocks, which is an essential steptoward more geologically realistic models.

3 A Crucial Starting Point: TheModel “Scaling”

The main advantage of laboratory modelling, i.e.that geological processes are simulated within thelimits of a laboratory and the working time of theresearcher, at the same time represents themethod’s biggest problem: the difference in scalebetween natural and laboratory volcanic plumb-ing systems. It is the scale gap between smallmodels and huge intrusions that often leads toscepticism among Earth scientists regarding theapplicability of laboratory models. Following thepioneering 19th century work that impressivelyreproduced natural structures, however, the 20thcentury saw a breakthrough with the develop-ment of the concept of scaling, with particularapplication to experimental models.

In 1937, Marion King Hubbert introduced the“theory of scale models” to the geological com-munity and thereby shifted phenomenologicalanalogue experiments from a qualitative to aquantitative approach. According to Hubbert(1937) and his followers (e.g. Ramberg 1970,1981), laboratory models should be geometri-cally, kinematically, and dynamically similar tothe natural system under investigation. Scaling istherefore an essential part for any laboratorystudy and needs to be considered to justify theapplicability of the model results. Nevertheless,


there is a lot of confusion among Earth scientistsabout correct scaling of laboratory models togeological systems, and first of all about themeaning of “scaling”.

The main confusion is related to the misun-derstanding of the goals of laboratory models.They are not designed to exactly mimic a givengeological system, as many Earth scientiststhink, but to understand underlying generic pro-cesses, either individually or in combination, andto identify or demonstrate physical laws gov-erning these processes. In order to prove that anyphysical laws identified in laboratory experi-ments apply to geological systems, these lawsshould be dimensionless, i.e. they are indepen-dent of the length scale, time scale, etc., which iswhy they are often called scaling laws.

The procedure for establishing dimensionlessscaling laws involves two steps: (i) a dimensionalanalysis of the considered physical problem, toidentify the dimensionless governing parameters,and (ii) a comparison of laboratory and geo-logical values of these dimensionless parametersto test the geological relevance of theexperimentally-derived physical laws.

3.1 Dimensional Analysis

The principles of dimensional analysis aredescribed in detail by e.g. Barenblatt (2003). Thefirst applications of such an approach in labora-tory experiments of volcanic processes has beenconducted by Tibaldi (1995) and Merle andBorgia (1996). The approach consists of identi-fying the dimensionless physical parameters thatgovern the processes to be addressed.

The first step is to list the n parameters withdimensions that are relevant for the processes to bestudied. In mechanical systems, usual dimensionsare those of length [L], mass [M] and time [T]. Ananalysis of thermo-mechanical systems requiresthe addition of the dimension of temperature [K].

Let us consider a simple conceptual example:the stability of a volcanic edifice. The aim of thefollowing paragraphs is only to provide anillustrative example of dimensional analysis. Inthe case of edifice stability, the effect of the

coefficient of friction is obvious, however it is notsuitable for illustrating the implementation ofdimensional analysis. Instead, we will considerthe effect of material cohesion on edifice stabil-ity, which has not been considered. In the fol-lowing sections, we will thus assume that thecoefficient of friction is taken as a constant, and itwill be ignored despite its established relevance.The dimensional analysis developed in the fol-lowing sections thus only aim at assessing theeffects cohesion on edifice stability.

The edifice has a height h ([h] = L), diameterD ([D] = L), a density ρ ([ρ] = M L−3), and thematerial making the edifice has a cohesionC ([C] = M L−1 T−2). The square brackets hererepresent the dimension, not the unit, of eachparameter. The volcano is exposed to gravita-tional acceleration g ([g] = L T−2). Again forillustrative purpose, we consider the coefficient offriction to be a dimensionless constant and wewill ignore it during the following analysis,despite its established relevance. From this list ofparameters, the experimentalist should separatethe governing parameters, i.e. those known andcontrolled, from the parameters to measure. Tostudy the stability of a volcanic edifice in thelaboratory, two approaches can be developed.A first approach (example 1) consists of control-ling both the height and diameter of the edifice,such that h, D, ρ, g and C are governing param-eters. The experimentalist builds volcanic edificesof controlled sizes using model materials ofvarying cohesions, and observes whether they arestable or not. A second approach (example 2)consists of controlling the height of the edificeonly, and measuring the diameter, such that h, ρ,g and C are the governing parameters and D themeasured quantity. The experimentalist graduallybuilds the cone with model materials of varyingcohesions at a critically stable slope to a givenheight h, and measures the diameterD of the cone,as in the experiments of Ramos et al. (2009). Webase the following sections on these examples.

The second step is to identify the number k ofthe governing parameters with independentdimensions. A set of parameters has independentdimensions if their dimensions cannot be


expressed as a function of each other’s (Bare-nblatt 2003). For instance, the height of a vol-canic edifice and the density of the rock haveindependent dimensions, because the dimensionof the density ([ρ] = M L−3) cannot be expressedas a function of the dimension of the depth([h] = L) only. In contrast, the rock cohesion([C] = M L−1 T−2), the density ([ρ] = M L−3), thegravity ([g] = L T−2) and the height of a volcanicedifice ([h] = L) do not have independentdimensions, given that the dimension of C is afunction of the dimensions of ρ, g and h:

C½ � ¼ M L�1 T�2 ¼ ½q� � ½g� � ½h�¼ M L�3� �� L T�2� �� Lð Þ ð1Þ

In example 1, it is thus possible to show thatρ, g, h are governing parameters with indepen-dent dimensions, whereas the dimensions ofC and D can be expressed as functions of thoseof ρ, g, and h. Note that the selection of thegoverning parameters with independent dimen-sions is not unique. For instance, one can chooseρ, g, and D as governing parameters with inde-pendent dimensions, C and h being the others.But in both cases, among the set of n = 5 gov-erning parameters, k = 3 parameters have inde-pendent dimensions. In example 2, there is onlyone possibility: ρ, g, and h are the governingparameters with independent dimensions, C isthe other governing parameter, and D is themeasured quantity. Therefore, among n = 4governing parameters, k = 3 parameters haveindependent dimensions.

The third step is to calculate the number m ofdimensionless parameters that characterise thephysical system to be simulated in the experi-ments. This number is easily calculated by usingthe Π-theorem (or Buckingham Π-theorem),which is the central theorem in dimensionalanalysis. It states that “a physical relationshipbetween some dimensional (generally speaking)quantity and several dimensional governingparameters can be rewritten as a relationshipbetween a dimensionless parameter and severaldimensionless products of the governingparameters; the number of dimensionless prod-ucts is equal to the total number of governing

parameters minus the number of governingparameters with independent dimensions”(Barenblatt 2003). This means that the numberm of dimensionless parameters to be defined isgiven by m = n − k.

In example 1, ignoring the coefficient of fric-tion, the total number of governing parametersbeing n = 5, and the number of governingparameters with independent dimensions beingk = 3, the number of dimensionless parametersgoverning the physics of a volcanic edifice ism = n − k = 2. If ρ, g and h are chosen as thegoverning parameters with independent dimen-sions, the two dimensionless parameters are:

P1 ¼ Cq� g� h ð2Þ

P2 ¼ hD ð3Þ

In example 2, again ignoring the coefficient offriction, n = 4 and k = 3, therefore the number ofdimensionless parameters governing the definedphysical system is m = 1. A clear choice for thisparameter is simply:

P1 ¼ Cq� g� h ð4Þ

In order to test the relevance of the dimen-sional analysis, the defined dimensionless num-bers should have a physical meaning. In example1, the physical meanings of Π1 and Π2 arestraightforward. Π2 expresses the slope of thevolcanic edifice. Π1 expresses whether the vol-canic edifice is gravity-controlled (Π1 → 0) orstrength-controlled (Π1 → ∞). In the first case,the material that composes the edifice is macro-scopically loose, such as sand, and the edifice isexpected to collapse under its own weight alongshear planes (i.e. faults). Conversely, in the sec-ond case the material is macroscopically cohe-sive, and the edifice is expected to be stable.

Several widely used dimensionless numbersare called by the name of their discoverer.A good example is the Reynolds number (Re),which quantifies the ratio between the inertial


forces in a flowing fluid and the viscous forces.A critical value of the Reynolds number sets theboundary between the laminar flow regime andthe turbulent flow regime. This number is thushighly relevant for magma flow. Another exam-ple is the Rayleigh number (Ra), which quanti-fies the ratio between the buoyancy forces of aheated fluid and the viscous forces (i.e. thoseforces resisting flow). A critical value of theRayleigh number sets the boundary between aconvective system and a non-convective system:this has high relevance for mantle convection andthe dynamics of magma chambers.

In example 2, we note that the measuredquantity D has not been used in the definition ofthe dimensionless parameter Π1. At this stage, wecan define a dimensionless parameter Π, which isdefined as a ratio between the measured quantityD and a function of the governing parameterswith independent dimensions. Here a definitionis given by Π = D/h. The Π-theorem implies thata dimensionless quantity Π to be measured in theexperimental study can be rewritten as a functionof the other m Π-numbers, such as:

P ¼ F P1;P2; . . .;Pmð Þ ð5Þ

The function F is the physical law that governsthe simulated processes. This relationshipbetween the dimensionless output and thedimensionless input parameters should dictate theexperimental strategy. In order to test the effectsof each dimensionless parameter Πi during anexperimental project, the dimensional experi-mental parameters should be varied such thatΠi issystematically varied, while the others are keptconstant. Applied to each Πi-number and so byconstraining the function F, the experimentalresults will contribute to deriving the physicallaws that govern the investigated processes.

This strategy can be adapted to two differentapproaches depending on the nature of the modeloutputs.

If the model output is not a measured quantity,but contrasting physical behaviours, the aim ofthe experimental procedure will be to explore theparameter space, i.e. to vary systematically thevalues of the dimensionless parametersΠi, to mapunder which conditions these contrasting physicalbehaviours occur (Fig. 5a). This procedure isequivalent to building a phase diagram, the con-trasting physical behaviours being physical pha-ses. Let us illustrate this approach with ourexample 1. An experimental procedure would

Π=

h/D

α

Π C/ρgh

α

Πh/

D

Π C /ρgh

Stable(a) (b)

Unstable

Fig. 5 Qualitative diagrams illustrating the two mainexperimental strategies defined from dimensional analy-sis. a Example 1, when the model output is not ameasured quantity but a number of physical behaviours,each behaviour (here stability or collapse of a volcanicedifice; see Sect. 3.1 for explanation) is plotted in adiagram with the dimensionless input parameters as x-and y-axes. This so-called phase diagram maps thephysical fields, in which the observed behaviours are

expected. The field transitions are commonly described bypower laws of the form P2 / Pa1. b Example 2, i.e. whenthe model output is a measured quantity Π (here the slopeΠ = h/D of a volcanic edifice at stability criterion; see textfor explanation), it is plotted as a function of thedimensionless input parameters (here Π1 = C/ρgh, seetext for explanation). The correlation between Π and Π1 isusually a power law of the form P / Pa1


consist of building a model volcanic edifice ofgiven and controlled height h and width D, withmaterials of given cohesions C (and a constantcoefficient of friction). Once the edifice is built,one can see if it is stable or unstable (collapses).The experimental strategy would be to run manyexperiments by varying systematically and inde-pendently Π1 and Π2 to explore how theseparameters control the stability of a volcanic edi-fice. By plotting all the experiments in a diagramof Π1 against Π2, and by indicating the corre-sponding physical behaviour with e.g. differentdata point symbols, it is possible to identifyphysical fields, or phases, separated by transitions.The transitions between the fields are mostlyexpected to be power laws of the form:P2 / Pa1.

If the model output is a measured quantity, theaim of the experimental procedure will be toestablish a correlation between the measureddimensionless parameter Π and the dimensionlessinput parametersΠi (Fig. 5b). In our example 2, anexperimental procedure would consist of gradu-ally building model volcanic edifices of givenheight h with materials of given cohesions C, andmeasure the final diameter D of the edifices. Theexperimental strategy would be to run manyexperiments by systematically varying Π1 toexplore how these parameters control the outputparameter Π. If the parameters Π and Πi are welldefined, they are expected to correlate, and mostlikely following a power law. Hence the correla-tion should appear as a straight line of slope α in alog-log plot, showing that the physical law linkingΠ and Πi is a power law of the form: P / Pa1.

3.2 Similarity to GeologicalSystems

After performing the dimensional analysis,identifying the Π-numbers, and obtaining theexperimental results, the geological relevance ofthe experimentally-defined scaling laws needs tobe tested. In other words, we need to test whetherthe processes simulated in the laboratory arephysically similar to the geological processes.This concept of physically similar phenomena iscentral to geological laboratory modelling. “Twosystems are considered similar if the values of

the dimensionless parameters are identical, evenif the values of the governing dimensionalparameters differ greatly” (Barenblatt 2003). Itmeans that although the scales of the laboratorymodels are drastically different to the scales ofthe geological systems they aim to simulate, thelaboratory models will be physically similar totheir geological equivalents if their respective Π-numbers have the same values. Therefore, theexperimentalist must compare the values of eachΠ-number in the laboratory with the values ofthese numbers in the geological system: if theranges of values overlap, the two systems aresimilar, and the experimental results are relevantto the geological system.

The principle of similarity can be also lessstrictly applied if we consider the physicalmeaning of the Π-numbers. Let us consider theheight-to-diameter ratio of a volcanic edificeΠ = h/D. In geological settings, Π is typically0.12–0.6 (Grosse et al. 2012). In laboratoryexperiments, the resulting values of Π will belarger or smaller than the geological values of Π,if the model material is too cohesive or too loose,respectively. In this case, the laboratory volcanicedifices are not rigorously similar to the geo-logical edifices. Nevertheless, in our example 2 ifa plot of the laboratory and geological values ofΠ against Π1 = C/ρgh shows an alignment alongthe same scaling law (Fig. 6), it would mean thatboth edifice types are governed by the samephysical law. Consequently, in both the labora-tory and geological systems, the physical regimesare the same, and the law identified in the labo-ratory model of volcano edifice can be consid-ered physically relevant to geological edifices. Inour example 1, it is also important to consider thephysical regime (i.e. the physical ‘phase’ in thephase diagram): if the geological edifice and thelaboratory edifices have different values of Π1and Π2, but if both are in the stable field, they canbe considered physically equivalent, withoutbeing similar sensu stricto. Likewise, the Rey-nolds numbers related to magma flow in thelaboratory and its geological prototype can bedifferent, but if both are way below the criticalvalue, the flow regimes are both laminar (seediscussion in, e.g. Galland et al. 2009).


4 Geological Applications

There is an extensive literature on laboratoryexperiments of volcanic plumbing systems. Westructure our review of these past studies accord-ing to the simulated geological structures or pro-cesses, which include: (1) dykes, (2) cone sheets,(3) sills, (4) laccoliths, (5) caldera-related struc-tures and intrusion, (6) ground deformation asso-ciated with shallow intrusions, (7) magma/faultinteractions and (8) explosive volcanic vents.

4.1 Dyke Formation

Dykes are the most common magma conduits inthe Earth. They are steeply inclined sheet-likeintrusive bodies that are discordant to anymechanical stratigraphy in their host rocks, andthat usually exhibit a very smallthickness-to-length ratio (between 10−4 and 10−2,Rubin 1995). The latter characteristic means thatmodel dykes in experiment set-ups of a fewdecimetres in length should theoretically be verythin (between 1 and 10−3 mm). This is verychallenging to achieve.

Magmatic dykes are commonly assumed to beemplaced by hydraulic fracturing (Pollard 1987;Lister and Kerr 1991). One of the first experi-ments that intended to simulate hydraulic frac-turing consisted of elastic gelatine as the modelrock, and mud as the model magma (Fig. 7)(Hubbert and Willis 1957). Although the moti-vation of these experiments was to simulatehydraulic fracturing in boreholes, the experi-mental results are relevant for igneous dykes andsills. These experiments demonstrated that thestress field applied to the model controlled theorientation of hydraulic fractures nucleating froma vertical pipe: the fracture planes were alwaysperpendicular to the least principal stress σ3, andparallel to the (σ1-σ2) plane (Sibson 2003). Inaddition, these experiments demonstrated quali-tatively the strong effect of mechanically layeredhost rocks on dyke propagation.

Experiments with gels have since been usedextensively to unravel various aspects of dykepropagation through the Earth’s crust. Gels aretransparent, so it is possible to track the evolutionof growing intrusions. This has proven verypractical for unravelling the physical processesgoverning dyke nucleation and propagation. Thefollowing provides a short summary of ourunderstanding of the physics of dykes from gelexperiments.

4.1.1 Dyke Nucleationfrom a Magma Reservoir

An important aspect of dyke formation is theirnucleation from a magma reservoir. Using gel+silicone experiments, Cañón-Tapia and Merle(2006) simulated dyke nucleation from an over-pressurised magma reservoir made of siliconeputty. The silicone putty was injected at a con-stant flow rate via a computer-controlled piston.The results suggest that dykes nucleate inresponse to the overpressure in the reservoir,rather than because of buoyancy forces arisingfrom the density difference between the siliconeand the gel. Interestingly, these experimentshighlight the discontinuous behaviour of dykepropagation from a magma reservoir, whichresults from a competition between the strainenergy accumulated in the gel and the pressureevolution in the reservoir.

α

Laboratoryvalues

Geologicalvalues

Π=

h/D

Π C/ρgh

Fig. 6 Schematic diagram illustrating a potential mis-match between laboratory and geological values of theslope h/D against C/ρgh for a volcanic edifice (see text forexplanation). The model and geological edifices exhibitdifferent values, implying that they are not strictly similar.Nevertheless, they plot on the same scaling law, showingthat they result from the same physical processes.Therefore, such diagram shows that the laboratory modelsare physically equivalent to the geological systems,without being strictly similar


Further to the above experiments, McLeodand Tait (1999) investigated the effects of magmaviscosity on dyke nucleation (Fig. 8a). By con-trolling the injection pressure rather than theinjection flow rate, McLeod and Tait (1999)showed that for an instantaneous increase inmodel magma pressure, the time delay requiredto nucleate a dyke from a reservoir in a gelatinehost increases with increased fluid viscosity(Fig. 8a). In addition, for gradually increasingpressure in the reservoir, the critical pressurerequired to nucleate a dyke is larger when themagma is more viscous. These results suggest

that the pressures required to form felsic dykesare much higher than for mafic dykes.

4.1.2 Dyke Propagationin a Homogeneousand Isotropic Medium—TheEffects of Buoyancy

Fundamental mechanisms controlling the propa-gation of dykes have also been studied by usingsimple gel experiments that take advantage of thegel’s isotropic and homogeneous nature. Thisnature means that effects on propagation that arerelated solely to buoyancy, i.e. the force arisingfrom the difference in density between themagma and its host rock, can be isolated andunderstood.

Takada (1990) concluded that the shapes andvelocities of propagating dykes depend largelyon the density difference between the intrudingfluid and the gel—i.e. that they arebuoyancy-controlled (Fig. 8b). In Takada’s(1990) experiments, the liquid was injected witha needle, either from the side or the bottom of themodels. Thus, the dykes were of controlled vol-umes, and neither the applied overpressure northe injection flow rate was controlled. Takada(1990) concluded that: (i) Intrusion shapes aregoverned by the balance between the buoyancyforces and the strength of the gel. If buoyancy issmall, the dykes are symmetrical penny-shaped,whereas if the buoyancy is large, the dykes arevertically asymmetrical in transverse sectionwhere they exhibit a tear drop shape with a fatcurved upper tip and a sharp straight lowertip. (ii) The propagation direction is governed bythe balance between the buoyancy forces and thestrength of the gel: if buoyancy is small, the dykepropagates in all directions, whereas if thebuoyancy is large, the dyke propagates domi-nantly upward. (iii) A buoyant crack of constantvolume propagates itself only if it is larger than acritical length (see also Algar et al. 2011).(iv) The propagation velocity of the dyke iscorrelated to the density difference between thegel and the injected fluid. (v) When buoyancyforces are higher, the leading edge of the intru-sion becomes broad, leading to the splitting ofthe dyke tip.

Fig. 7 Characteristic results of pioneering experiments ofHubbert and Willis (1957) to address the mechanics ofhydraulic fracturing. a Drawing of experimental setupwith anisotropic horizontal stress. A cylinder of gelatine iscompressed in one direction between two plates. Mud isinjected in the gelatine, and form hydraulic fractures.b Photograph of vertical hydraulic fracture obtained fromexperimental setup displayed in a. The fracture isperpendicular to the least principal stress, which ishorizontal and parallel to the plates. c Drawing ofexperimental setup with isotropic horizontal stress. A cyl-inder of gelatine is compressed horizontally in everydirection by a series of elastic strings. d Photograph ofhorizontal hydraulic fracture obtained from experimentalsetup displayed in c. The fracture is perpendicular to theleast principal stress, which is vertical


In addition to examining buoyancy on dykepropagation, Menand and Tait (2002) investi-gated the effect of overpressure in a magmareservoir. They imposed an overpressure byplacing a water reservoir at given vertical dis-tances above the model. They show that dykepropagation exhibits two regimes: (i) an initialoverpressure-dominated regime during which thedyke propagates both vertically and laterally, and(ii) a subsequent buoyancy-dominated regimeduring which the dyke propagates mostly verti-cally. The transition between both regimes iscontrolled by a critical size of the dyke, inagreement with Takada’s (1990) experiments.

Finally, buoyancy-driven cracks of constantvolume in gelatine experiments accelerate as they

approach the free surface (Rivalta and Dahm2006). Such a phenomenon has also beenobserved in volcanoes, such as Piton de la Four-naise, Réunion Island (Battaglia 2001), andappears crucial for forecasting the timing of vol-canic eruptions by means of dyke-induced seis-micity (Rivalta andDahm 2006). Interestingly, theexperiments show that the free surface effects arelarger when the fracture is ascending more slowly(i.e. when the buoyancy forces are smaller).

4.1.3 Effects of Contrasting MagmaViscosities on CompositeDyke Propagation

Magma batches of different compositions, and soof different viscosities, can lead to the formation

(a) (b)

Fig. 8 a Schematic drawing of experimental setup(upper left) and photograph of a characteristic experi-ments of dyke nucleation from a magma reservoir (upperright; McLeod and Tait 1999), in which fluids of variousviscosities are injected at controlled pressures in a gelatinemodel from a cavity. The lower left graph shows how theviscosity controls the fluid pressure required to nucleate adyke, whereas the lower right graph shows how theviscosity of the injected fluid controls the time scale of

dyke nucleation from a reservoir. b Experimental setup(left) of Takada’s (1990) experimental study of dykepropagation. Liquids of various densities and viscositieswere injected via a syringe, and their subsequent prop-agation was monitored with photographs. The series ofphotographs show profile (left) and front (right) views ofthe time evolution of a propagating dyke during acharacteristic experiment


of composite dykes, the cores of which are felsicand the rims are mafic (Walker and Skelhorn1966). A spectacular example is the Streitishorndyke, Eastern Iceland (Paquet et al. 2007).Koyaguchi and Takada (1994) simulated thepropagation of a dyke from a magma reservoircontaining two fluids of contrasting viscositieswithin an isotropic gelatine medium. They showthat low viscosity fluid (mafic magma) at thedyke tip controls the tip’s propagation. Thislower viscosity fluid in the dyke tip lubricates thesubsequent flow of the higher viscosity fluid(felsic magma) into the middle of the dyke. Suchprocess may considerably enhance the drainageof a silicic magma chamber and the flow ofviscous felsic magmas over large distances.

4.1.4 Effects of MechanicalHeterogeneity (PrexistingDykes or Layering) on DykePropagation

The medium through which dykes propagate innature, the Earth’s lithosphere, is neither isotro-pic nor homogeneous. Two first-order planarheterogeneities that can greatly influence dykepropagation include pre-existing dykes and sed-imentary layering. Experiments investigating theeffects of such heterogeneities (Maaløe 1987;Bons et al. 2001; Rivalta et al. 2005; Kavanaghet al. 2006) show that they represent fundamentalfeatures controlling the average dyke propagationvelocity, the propagation direction, and possiblyalso dyke geometry.

Maaløe (1987) and Bons et al. (2001) con-ducted gelatin and air/water model experimentsthat suggest that dykes propagate in a stepwisemanner due to successive magma batches.Maaløe (1987) showed that an initial magmabatch forms a dyke, which stops after some dis-tance. This initial dyke only propagates further ifa new magma batch reaches it. Bons et al. (2001)simulated dyke propagation in heterogeneousgelatine, and showed that the path of an initialdyke greatly controls the propagation of sub-sequent dykes. In addition, Bons et al. (2001)show that local heterogeneities stopped a propa-gating dyke, which only continues propagating ifit is fed by a new magma batch.

Rivalta et al. (2005) and Kavanagh et al.(2006) addressed the mechanisms of dyke prop-agation through mechanically-layered media,which are common in sedimentary basins andlava piles. They show that the velocity of thepropagating dyke is correlated with the strengthand Young’s modulus of the host: a dyke decel-erates when it reaches a stronger layer, and viceversa. In addition, the nature of the layer interfacecan considerably affect the 3-dimensional shapeof a dyke (Fig. 9). For instance, if a dyke reaches avery strong layer interface, its vertical propaga-tion is arrested and it propagates laterally. If thedriving pressure in the dyke is large enough, thedykes can turn into horizontal sills at such inter-faces (see also the Sect. 4.3 below).

4.1.5 Effects of Anisotropic StressFields (From RegionalTectonics or LocalTopography) on DykePropagation

It has been long hypothesised from field rela-tionships that dyke emplacement is greatly con-trolled by the regional stress field (Hubbert andWillis 1957; Sibson 2003; Takada 1999). If thestress field is homogeneous but anisotropic (i.e.,the magnitudes of the principal stresses differ),dykes are expected to be parallel to each other,and perpendicular to the least principal stress, σ3(Anderson 1936). In addition, topographic fea-tures, such as volcanic edifices, are thought toproduce local anomalous stress fields that cangreatly affect the propagation of dykes (Odé1957; Johnson 1970; Nakamura 1977). Thiseffect of gravitational stresses related to topog-raphy on dyke geometry and propagation hasbeen subject of numerous experimental studies.

Gelatine experiments have been used toinvestigate how the stress generated by the loadof a volcanic edifice affects dyke propagation inthe region around the edifice. Muller et al. (2001)and Watanabe et al. (2002) used injected dykes atvarying distances from a load, and observed thatdykes close to the load were attracted toward it,whereas dykes far away from the load were not(Fig. 10). They show that the critical distance atwhich a dyke is affected by the load is correlated


(Rivalta et al., 2005)

(a)

(b)

Fig. 9 a Experimental apparatus designed by Rivaltaet al. (2005) to study dyke propagation through layeredmedia. The model magma is air, and the model rockconsists of two layers of gelatine of different concentra-tions. b Time series of photographs of a characteristicexperiment of dyke propagation in a layered medium.Here the upper layer is more resistant than the lower layer.

When the rising dyke reaches the interface between thelayers, it is temporarily stopped upward but propagateslaterally (photographs bR, cR, dR), until the dyke tipstarts piercing through the upper, tougher layer (photo-graph eR). Interestingly, the dyke is wider in the upperlayer than in the lower layer, although it has the samevolume


with the ratio of average pressure at the base ofthe load (i.e. the height of the edifice) to the dykedriving pressure: larger loads attract more dis-tanced dykes. The particular trajectories taken byattracted dykes are also affected by this ratio.This attraction of dykes rising through the crustby volcanic loads may help to explain theabsence of volcanism between large volcanoes.

Laboratory experiments of dyke emplacementalso show that topography-derived stresses cangreatly control the orientation of dykes withinvolcanic edifices (Fiske and Jackson 1972).Gelatine experiments by Hyndman and Alt(1987), McGuire and Pullen (1989), Delcampet al. (2012b), for instance, show that dykesorientate along the long axis of an elongatedvolcanic edifice, albeit with some divergence atthe edifice periphery (Tibaldi et al. 2014). Inaddition, the experiments of McGuire and Pullen(1989) show that: (i) shallow dykes below edi-fices stop propagating upward, and start

propagating laterally to give rise to a lateral fis-sure eruption, and (ii) the behaviours of intrudingdykes differ according to their initial positionwith respect to the summit of the edifice. Kervynet al. (2009) show a similar deflection of the dykeaway from the summit of an edifice standing on abrittle substratum of controlled thickness(Fig. 11a); deflection depends on the thickness ofthe substratum, on the edifice slope, and on theoverpressure within the dyke. These experimen-tal results are consistent with lateral dyke prop-agation observed in many volcanoes, andcorroborate well the theoretical analyses of Pineland Jaupart (2000, 2004).

Rapidly constructed topographic loads likevolcanoes can be unstable and sectors of anedifice can tend toward collapse. Walter andTroll (2003) used gelatine experiments to inves-tigate the effect of a volcano sector instability onthe formation of dyke swarms (Fig. 11b). At thebase of a gelatine edifice, they simulated a

(Muller et al., 2001)

Fig. 10 Experimental study of Muller et al. (2001), inwhich air-filled dykes are injected in a gelatine model, atthe surface of which a load is placed. Dyke injection issuccessively performed at an increased lateral distance xfrom the position of the load. The upper right photographshows dyke trajectories during an experiment: the dykecloser to a critical distance xc are attracted toward theload, whereas the other dykes are not attracted by the

load. The lower right diagram shows that the criticaldistance xc in the experiments (black discs) is a linearfunction of the ratio between the pressure induced by theload (Pload) and the excess pressure in the dyke (ΔPm).Muller et al. (2001) also performed numerical modelling,and obtained different behaviours of xc as a function ofPload/ΔPm (see results in lower right graph.)


(Walter and Troll, 2003)

(Kervyn et al., 2009)(a)

(b)


localised décollement, above which the sector ofthe edifice was considered unstable. Theyshowed that: (i) injection into the stable sector ofthe edifice results in two main radial dykeswarms, (ii) injection into the unstable sector ofthe edifice results in swarms reflecting the extentof the unstable sector, and (iii) injection close tothe transition between the stable and the unstablesectors mainly produce three swarms, two ofwhich follow the stable/unstable discontinuity.These results corroborate geological observationsfrom volcanic centres exhibiting triaxial riftzones, such as Tenerife, Canary Islands (e.g.Delcamp et al. 2012a).

Sector instability and sliding have been linkedwith active intrusion or the presence of intrusivecomplexes at several volcanoes. Models studyingthe interaction between gravitational instabilityand intrusions have been conducted by Mathieuand van Wyk de Vries (2009), Delcamp et al.(2012b), and by Norini and Acocella (2011).Mathieu and van Wyk de Vries (2009) modelledthe development of the Mull centre in Scotland,showing how large intrusions could developdykes along gravity slides. Delcamp et al.(2012b) found that dykes would propagate out ofa central intrusion into rift zones and appliedtheir models to the distribution of inclined dykesand intrusive bodies on La Reunion Island,France. Norini and Acocella (2011) used acombination of high viscosity silicone and lowviscosity oils to study the relationship betweenmagma intrusion, gravity sliding and tectonicextension at Mt. Etna, Italy. They conclude that

the magmatic forcing and gravity effects greatlyoutweigh the regional tectonics in controllingintrusion and instability on Mt. Etna.

4.1.6 Interaction of CoevallyPropagating Dykes

Laboratory experiments enable the emplacementof several coeval dykes to see how they interact. Itis expected that the stress field induced by prop-agation of one dyke would influence the propa-gation of another (Delaney and Pollard 1981).Takada (1994a, b) carried out gelatine experi-ments, in which he injected two coeval dykes.The experiments show that: (i) two propagatingliquid-filled cracks are likely to coalesce; (ii) apropagating liquid-filled crack is unlikely tocoalesce with a nearby solidified crack; (iii) largeregional differential stresses impede the coales-cence of magma-filled cracks, and control theformation of parallel dykes; (iv) a large magmasupply rate can produce a complete stress fieldrearrangement that overcomes the regional stressfield, leading to the formation of radial dykes, asobserved in many exhumed volcano plumbingsystems (Odé 1957; Nakamura 1977).

In other gelatine experiments, Ito and Martel(2002) tested the effect of the critical dykespacing xc on dyke interaction: if the distancex between the dykes is smaller than xc, the dykescoalesce, whereas if x > xc, the dykes do notcoalesce (Fig. 12). Also, increasing the regionaldifferential stress reduces the value of xc. Theexperiments show that when regional differentialstresses are small, xc scales with only a few times

Fig. 11 a Experimental study of Kervyn et al. (2009), inwhich an water-filled or an air-filled dyke was injectedinto a gelatine model, at the surface of which was avolcanic edifice made of sand. Left Drawing of experi-mental setup. Centre Time series photographs of acharacteristic experiment of dyke rising under a volcanicedifice. Right Evolution of the dyke outline illustratingthat the dyke rise velocity decreases when approachingthe cone base. b Experimental study of Walter and Troll(2003), in which dyed water was injected into a volcanicedifice made of gelatine. Top Geometry of experimentalcones in stable situation (A) and unstable situation (B).Due to gravity, the cones spread outward, partially slidingon a basal lubricant. In the unstable situation, only oneflank was lubricated. IP injection point (10 mm above

base), basal radii R3 > R2 > R1; cone height h > h2 > h3.Through small holes drilled in the basal plate, injectionwas possible at various positions. Bottom Summarizedarrangement of hydro-fractures propagating in locallydestabilized edifices relative to injection point andeccentricity of a creeping sector. A Injection into stablesector. B Injection into creeping sector. C Injection closeto the interface stable/unstable sector. Stereoplots illus-trate the statistic orientation of fractures: each black dot inpole plots refers to the distal locality of an experimentalfracture, measured in azimuth and distance from the pointof injection. In frequency–azimuth (rose) diagrams (polarlines, sector size = 8°), the length of each rose sector isproportional to the frequency of orientation that lieswithin that sector (Walter and Troll 2003)

b


the dyke height, i.e. dyke interaction can poten-tially focus magma transport over large verticaldistances. Such a process can help explain thecritical distance observed between volcanoes atthe Earth’s surface, even though these volcanoesare fed from broad melting zones in the mantle.

4.1.7 Inelastic ProcessesAssociated with DykePropagation

With gelatine experiments, one assumes that themodel host rock behaves as a nearly purelyelastic solid, and that dykes propagate accordingto the Linear Elastic Fracture Mechanics (LEFM)theory. Nevertheless, field observations (Mathieuet al. 2008; Kavanagh and Sparks 2011; Daniels

et al. 2012) suggest that host rock behaviourduring dyke emplacement is substantiallyinelastic. Such inelastic processes can beaddressed in laboratory models made of cohesivegranular materials.

The experiments of Mathieu et al. (2008) andKervyn et al. (2009), which use ignimbritepowder and Golden Syrup, show qualitativelythat shear bands form at the tips of propagatingdykes, leading to the splitting of the dyke tips toform V-shaped (cup-shaped) intrusions. Suchdyke tip splitting has also been observed in the3D experiments of Galland et al. (2009) andGalland (2012), which used silica flour andvegetable oil. These results suggest that dykesmay propagate as viscous indenters, where the

Fig. 12 a Schematic drawing of the experimental setupof Ito and Martel (2002) for studying dyke interactions.b Time series of photographs of a characteristic

experiment illustrating the interaction and coalescenceof two propagating parallel dykes


viscous material penetrates the cohesive granularmaterial like a chisel pushed into plaster or aspade pushed into the ground (Donnadieu andMerle 1998), rather than as a simple mode Ifracture, particularly if the host rock cohesion isrelatively low. Quantitative 2D experimentalresults of Abdelmalak et al. (2012) (Fig. 13a)further show that: (i) small-scale reverse shearbands form at the vicinity of the propagatingdyke tip, and (ii) the model surface lifts up due todyke emplacement (Fig. 13b). These two obser-vations are incompatible with the LEFM theory,and the results suggest that models made ofcohesive granular materials may provide impor-tant insights into the complex mechanics gov-erning dyke emplacement in low strength rockmasses. Given that these models have beendesigned only recently, they offer broad possi-bilities for future experimental studies of dykeemplacement.

4.1.8 Effects of Cooling on DykePropagation

A critical aspect of magma intrusion is the effectof cooling, which is usually neglected in labo-ratory experiments (e.g., Galland et al. 2009),because it is technically challenging to control.Taisne and Tait (2011) performed experiments inwhich they injected a wax into a model made ofgelatine; the gelatine temperature was lower thanthe solidus temperature of the wax. The tem-perature difference between the injected wax andthe host gelatine was varied, as was the balancebetween the heat injected into the system and theheat diffusing from the wax into the gelatine host.Three main behaviours were observed: (i) whenthe injected wax temperature was large comparedto its solidus temperature, and when the heatinflux was large compared to the diffusive heatloss, the dyke propagated continuously, like anon-solidifying dyke; (ii) when the injected waxtemperature was close to its solidus temperature,and when the heat influx was small compared tothe diffusive heat loss, the dyke did not propa-gate; (iii) when the wax temperature and the heatfluxes lay between the above end members, thedyke propagates in a stepwise, intermittent,

manner due to local clogging of the dyketip. This intermittent behaviour may help explainthe occurrence of seismic bursts recorded duringdyke emplacement in volcanoes (Hayashi andMorita 2003; White et al. 2011) as a consequenceof the thermo-mechanical interaction of the dykeand its host rocks.

4.2 Cone Sheet Emplacement

Cone sheets are prominent features in manyvolcanoes on Earth. In eroded volcanoes, thesesheet-like intrusions strike concentrically aboutand dip in toward the volcano centre, and theycommonly occur as dense swarms. Like dykes,they are typically discordant to any mechanicalstratigraphy in their host rocks. A classic exam-ple of a cone sheet swarm is found in the Ard-namurchan intrusive complex, NW Scotland(Richey et al. 1930; O’Driscoll et al. 2006;Burchardt et al. 2013). Other good examples arefound in the Canary Islands (Ancochea et al.2003), Galápagos Islands (Chadwick and How-ard 1991; Chadwick and Dieterich 1995), andIceland (e.g. Schirnick et al. 1999; Walker 1999;Klausen 2004; Burchardt et al. 2011).

Despite the prominence of cone sheets, little isknown about their much-debated emplacementmechanisms (e.g. Phillips 1974; Klausen 2004;Burchardt et al. 2013). One reason is that labo-ratory cone sheets rarely occur in gelatineexperiments, which mostly simulate either dykes(Takada 1990; Lister and Kerr 1991; Dahm2000; Rivalta and Dahm 2006; Le Corvec et al.2

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