The growth of dykes from magma chambers

Journal of Volcanology and Geothermal Research 92 Ž1999. 231–245 www.elsevier.comrlocaterjvolgeores

The growth of dykes from magma chambers Paul McLeod ) , Stephen Tait Laboratoire de Dynamique des Systemes Geologiques, Institut de Physique du Globe de Paris, 4 place Jussieu-Boite 89, 75252 Paris cedex 05, France Accepted 31 March 1999

Abstract Failure of the margins of magma chambers produces dykes which may breach the ground surface and cause an eruption. The pressurization and failure of liquid-filled cavities is investigated with experiments in which gelatine is used as the analogue for a crustal medium. A theoretical analysis is derived which considers dyke nucleation to occur by the failure of a pre-existing magma-filled crack in the chamber wall. Flow of magma from the chamber into the crack gradually pressurizes and widens the crack. Eventually, the stress conditions for failure of the crack-tip are attained and dyke propagation commences. Magma viscosity influences the flux into the crack and hence the rate at which the crack-pressure increases. Higher viscosity magmas consequently require greater chamber overpressures andror longer time delays before dyke nucleation occurs. The theoretical analysis is in approximate agreement with experimental results and provides estimates of the critical conditions required for a magma chamber to rupture. If a basaltic magma chamber is suddenly pressurized Že.g., by replenishment., the delay before dyke nucleation is predicted to be on the order of hours to days. For rhyolitic magmas, the delays are typically many years. Where chambers are gradually becoming pressurized Že.g., by exsolution during magmatic crystallization., cracks in the walls of relatively silicic magma chambers are likely to be further from pressure equilibration with the chamber, resulting in the development of higher chamber overpressures before dyke nucleation occurs. Solidification in silicic cracks which are slowly pressurizing may inhibit crack-tip failure; greater chamber overpressures are consequently required before dykes eventually form. The implications of higher chamber overpressures during dyke nucleation are that repose periods are longer, dykes propagate further before freezing, which allows dykes to transport silicic magmas over substantial distances, and a more vigorous eruption occurs if the dyke breaches the surface. q 1999 Elsevier Science B.V. All rights reserved. Keywords: dyke; crack; magma chamber; overpressure; deformation; failure; viscosity

1. Introduction Dyke flow is the dominant mechanism of crustal transport for mafic magmas and possibly also for

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more silicic varieties ŽPetford et al., 1993, 1994; Rubin, 1993a.. Previous studies focused on the growth of established dykes. Results include the conclusion that for dykes over a few metres in length, propagation rates are dominantly controlled by viscous drag of the magma, with the resistance of the host rock to failure being relatively unimportant ŽSpence and Turcotte, 1985; Emerman et al., 1986;

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Lister and Kerr, 1991.. Propagation continues until the dyke breaches the surface in an eruption or ceases motion whilst still entirely within the crust, due to either the driving-pressure becoming sufficiently depleted or solidification closing the dyke faster than it can grow ŽDelaney, 1987; Bruce and Huppert, 1990; Rubin, 1993b, 1995a.. Dyke initiation has received relatively little attention. Sleep Ž1988. investigated the origin of dykes from partly molten source regions in the mantle. The generation of dykes from crustal magma chambers will be considered here. The pressure within a magma chamber can increase rapidly when there is a sudden input of fresh magma from a deeper source ŽSparks et al., 1977.. Alternatively, the pressure can grow slowly due to the build-up of exsolved gases during crystallization ŽBlake, 1984; Tait et al., 1989.. Increasing chamber pressures causes the chamber to inflate, the associated deformation of the surroundings generates seismic activity and ground surface deformations that are often detected as the precursor to eruptive activity Že.g., Johnson, 1987.. With sufficient pressurization, the chamber walls eventually fracture and a dyke is produced. Previous studies of the deformation around magma chambers assumed that the chamber walls fracture when they are exposed to tensile stresses greater than the tensile strength of the wall rocks ŽBlake, 1981; Sammis and Julian, 1987; Tait et al., 1989; Sartoris et al., 1990; Parfitt et al., 1993.. In this study, we consider in detail the mechanics of the process of chamber failure and dyke nucleation. Laboratory experiments using analogue materials provide a method of observing the process directly. Based on the results of these experiments, a theoretical model of dyke nucleation is developed and then applied to magmatic systems. The mechanics of the failure process determines the maximum chamber pressure attained, which will also be the pressure driving dyke propagation and eruption. An understanding of the failure criteria, combined with assessments of chamber conditions from geophysical data, will improve predictions of volcanic behaviour. The mechanism of dyke nucleation is also significant in determining the manner of magma transport and the mechanics of pluton emplacement.

2. Experimental technique A solidified aqueous gelatine solution is considered an appropriate crustal analogue because it is capable of initial elastic deformation followed by brittle failure. It is also conveniently transparent, and has been employed in several previous studies of dyke propagation ŽFiske and Jackson, 1972; Hyndman and Alt, 1987; Maaloe, 1987; McGuire and Pullen, 1989.. Twelve percent by weight of gelatine powder was dissolved in distilled water during heating. A 0.1%-sodium hypochlorite was added to prevent fungal growth. The elastic stiffness of the gelatine Žs elastic shear modulusrŽ1 y Poisson’s ratio.. was measured as 13 " 1 kPa. The experiments were conducted in a vertical cylindrical perspex tank of 29 cm internal diameter and 40 cm height ŽFig. 1.. The cavity representing the magma chamber and its feeding conduit was created using a mould. This consisted of an inflatable elastic balloon threaded through a metal tube; these were inserted into the tank through the floor and then the balloon was inflated to create a spherical form at the top of the tube. The gelatine solution was poured into the tank and left to cool and set for 48 h. During this period, the top of the tank was sealed to prevent evaporation of moisture from the gelatine. The balloon was then deflated and removed along with the

Fig. 1. Diagram of the experimental apparatus.

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metal tube, care being taken to ensure no tears or fractures formed on the cavity walls during removal. Initial chamber dimensions were a horizontal diameter of ; 4 cm and a vertical diameter of ; 3.5 cm. The total depth of gelatine was ; 22 cm and the depth to the top of the chamber was from 4.5 to 8 cm. The conduit had a diameter of ; 0.8 cm. Measurement of the gelatine surface and chamber dimensions during the experiments was achieved by aligning scales on either side of the tank, and by accounting for the distortion due to the curved surface of the tank. Prior to commencing the experiments, the chamber surface was extensively cut. The cuts were both vertical and horizontal Ži.e., along lines of longitude and latitude. and to uniform depths of ; 2 mm. They were obtained using a needle of adjustable orientation mounted on a rod. Magma was represented by dyed water, dyed aqueous hydroxyethyl cellulose ŽNatrosol. polymer solutions and silicon oils of various viscosities. Maximum errors on the viscosity measurements are ; 25%. The Natrosol solutions are miscible with gelatine whilst the oils were immiscible; however, this difference had no noticeable influence on the results. Hydration of the gelatine during the aqueous experiments was sufficiently slow to have no impact on the mechanical properties of the gelatine. The pressure of the liquid in the chamber was measured from the relative height of the external reservoir ŽFig. 1.. Values of the chamber pressures will be given with respect to the stress state of the surrounding gelatine Žinitially lithostatic., i.e., as underpressures or overpressures. The exact value varied slightly with height within the chamber and conduit due to small density contrasts between the liquids and gelatine; the initial centre of the chamber will be taken as the reference level. Two sets of experiments will be described. In the first set of experiments dyed water, dyed Natrosol solutions and silicon oils were used as the magma analogue. At the start of each experiment, the chamber was at a small underpressure. Then, the chamber pressure was progressively increased in steps. Each new step was postponed until flow out of the external reservoir appeared to have ceased. This method of gradually increasing the chamber pressure was continued until the chamber walls ruptured and a dyke formed. In the first couple of experiments,

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there was no cutting of the chamber surface prior to commencing the experiment. In all other cases, in both the first and second sets of experiments, the cutting was performed as described above. In the second set of experiment, silicon oils were used as the magma analogue. These experiments involved a more sudden variation in chamber pressure. The pressure was initially raised to a small overpressure of 15 Pa, and was then allowed to equilibrate for 24 h. After the equilibration period, the chamber overpressure was raised to 1200 Pa in a single step, and then the time until a dyke formed was measured. The experiment with the highest viscosity liquid was performed under slightly different conditions, the initial overpressure was 24 Pa and only 5 h was allowed for equilibration.

3. Experimental observations 3.1. First set of experiments — progressiÕe pressurization In the initial experiments, where there had been no pre-cutting of the chamber surface, the increasing

Fig. 2. Dimensionless vertical and horizontal chamber diameters Ždiameters divided by the diameter at zero overpressure. as a function of chamber overpressure for the experiment where the liquid was aqueous Natrosol solution of viscosity 0.025 Pa s. Maximum errors on measurements are "1 mm, which corresponds to "0.025 on the normalised values.

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Fig. 3. A sequence of photographs showing the development of an experimental dyke. The horizontal diameter of the chamber is ; 4 cm. Ža. Side view, initial stage of dyke growth. A small dyke has formed on the left side of the chamber. Žb. Side view, the dyke has now propagated further into the gelatine. Žc. Plan view, the photograph was taken at approximately the same time as the image in Žb.. Žd. Side view, the dyke has grown further and now encircles the chamber. The variation in dyke dip results in a disc-cone hybrid morphology. Že. Plan view, the photograph was taken at approximately the same time as Žd..

chamber overpressures eventually resulted in the formation of a dyke which nucleated on small corrugations in the cavity wall immediately around the join between the chamber and conduit. These corrugations in the chamber surface were caused by folds in the balloon material where it had entered the metal tube, and thus were an experimental artefact. The cuts in the chamber surface in all the other experiments provide equally suitable nucleation sites all around the chamber. These experiments will now be described in detail. With increasing overpressures, the chamber and conduit inflated. Typical variations in chamber dimensions are shown in Fig. 2. The dimensionless

Fig. 3 Žcontinued..

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Fig. 3 Žcontinued..

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diameter to overpressure relationship was the same, within experimental error, for all experiments, indicating the consistency of the elastic response of the gelatine. The increase in conduit diameter was between 1 to 3 mm Žfor the low and high viscosity liquid experiments, respectively.. The inflation induced elastic deformation of the gelatine; this was dominantly stretching deformation with a small component of compression. The surface expression of deformation was a broad low dome, which attained maximum heights of less than 0.5 mm. Once there was sufficient overpressure, a dyke would nucleate on one of the pre-cuts in the chamber surface just above the equator of the chamber ŽFig. 3.. The dykes had curved fronts, which produced disc-shaped dyke geometries, or partial discs. In the initial stages, the dyke had an approximately hemicircular form against the chamber surface. With further growth, the dyke would encircle the chamber and acquire the shape of a complete disc; in some instances, dyke propagation was halted by eruption before this occurred. In a couple of experiments, small subsidiary dykes formed at other locations on the chamber, these were relatively minor and growth soon ceased whilst the main dyke continued to propagate. Maximum dyke thicknesses were approximately ; 2 mm at the dyke entrance. Average dips of the main dykes were 158–288. The dyke was not a completely flat disc with uniform dip as there was a limited tendency for the local dyke dip to slope towards the chamber centre, so that parts of the disc were twisted to create a dyke form which was a hybrid between a flat disc and a cone with gently dipping sides. Propagation rates were approximately 1–3 cmrmin when dyed water formed the dyke, and were slower for more viscous liquids. An asymmetric mound formed at the gelatine surface as the dyke approached to within ; 2 cm depth. This attained a maximum height just prior to eruption of ; 5 mm. The dyke intersected the gelatine surface at the bottom of the steepest slope of the mound, with the gentler slope being above the deeper sections of the dyke. An arcuate fissure in the gelatine surface was produced, though which the liquid erupted. The fissures were 5–7 cm long, the curvature resulted from the slightly conical dyke form. Eruption reduced the chamber to zero overpressure. The chamber and

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conduit returned to their earlier dimensions, the surface mound flattened and the dyke fracture closed leaving a very thin layer of liquid trapped between the dyke walls. In the experiments with higher viscosity liquids, larger chamber overpressures were attained before dyke nucleation ŽFig. 4.. There was correspondingly more deformation of the gelatine. In the two lowest viscosity liquid experiments, the average dimensionless chamber diameter at failure was 1.07 while for the two highest viscosity liquid experiments the average dimensionless diameter was 1.12. 3.2. Second set of experiments — sudden pressurization Fig. 5 shows the measured time intervals between the moment when the chamber overpressure was increased to 1200 Pa and the point at which a dyke was first identified. For approximately the first third of this time interval, gradual inflation of the chamber was observed. This indicates that the increase in chamber pressure was not instantaneous, probably as a result of viscous drag in the pipe between the reservoir and the chamber slowing the flux of liquid into the chamber. Initial dyke growth was very slow, with rates increasing as the dyke lengthened. Other aspects of the growth and geometry of the dyke were similar to the description above.

Fig. 4. Chamber overpressures at the point of dyke nucleation for liquids of various viscosities Žfirst set of experiments..

Fig. 5. Experimental measurements and theoretical estimates of the time delay until a dyke formed following a sudden chamber pressure increase, as a function of liquid viscosity Žsecond set of experiments..

3.3. QualitatiÕe interpretation of experimental obserÕations 3.3.1. Dyke geometry Dyke propagation is perpendicular to the least compressive stress in the surrounding medium. Tangential stresses Žhoop-stresses. around a pressurized spherical cavity in an infinite elastic medium are equal for all positions around the cavity; i.e., there is no preferential dyke propagation direction. The preferred dyke orientation in the experiments reflects the perturbation of the stress regime due to the gelatine Žfree. surface ŽAnderson, 1936; Robson and Barr, 1964; McTigue, 1987.. The tendency of the dyke to become partly twisted Ži.e., to form a hybrid disc-cone morphology. was a result of the symmetry of the stress field around a vertical axis. Perturbations to the stress field caused by the growing dyke itself may have prevented the dyke from developing into a perfectly symmetric cone. Differing and more intricate dyke morphologies may occur in response to crustal stress fields. 3.3.2. Dyke nucleation The influence of liquid viscosity on the chamber overpressure acquired before dyke growth occurred Žin the first set of experiments. and the time required for dyke nucleation after a sudden increase in chamber pressure Žin the second set. demonstrates the

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importance of liquid flow during the dyke nucleation process. Our interpretation is that the pre-cut cracks in the chamber wall must pressurize to the required crack overpressure before propagation of the crack-tip can occur. Cracks are pressurized by influx of liquid from the chamber and the flow rate would be expected to be inversely proportional to viscosity. In the second set of experiments, the slower rate of crack pressurization for the more viscous liquids resulted in longer time delays before a dyke formed ŽFig. 5.. The variation in the apparent critical chamber overpressures in the first set of experiments ŽFig. 4. is due to differences in the degree of equilibrium between the cracks and the chamber at the point of failure. Each successive increment in chamber pressures occurred when flow out of the reservoir had appeared to have ceased. However, infiltration into the cracks was still continuing but was not noticed due to the very small liquid quantities involved. In the experiments with the lowest viscosity liquids, the cracks are likely to have been nearly fully equilibrated with the chamber before each successive increase in chamber pressure occurred. However, for the higher viscosity liquids, crack pressurization was much slower and probably incomplete when the next rise in chamber pressure took place. Consequently, in the higher viscosity liquid experiments chamber pressures were at larger values when failure eventually occurred. The crack overpressure required to cause propagation of the tip will have been the same in all experiments. If the high viscosity liquid experiments had been allowed sufficient time to full equilibrate between each incremental increase in chamber pressure, then the critical chamber overpressure would have also been the same in all cases.

4. Theoretical analysis An analysis is now presented of the dynamics of dyke nucleation. Ductile deformation will be neglected, consequently the analysis is best suited to magmatic activity in upper to mid crustal levels. We consider a spherical magma chamber which is contained within a homogenous elastic crust, and assume that the remote crustal stress field is hydrostatic and equal to the lithostatic load, which is appropriate where the effects of tectonic forces and

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the influence of the free surface are negligible ŽMcGarr, 1988.. An increase in magma chamber pressure leads to chamber inflation and perturbs the stress field in the surrounding crust. The stress tangential to spherical surfaces concentric with the chamber Žhoop-stress. is given by ŽSammis and Julian, 1987.:

st s s 1 q

1 2

ž

1y

Pc

s

rql

/ž / r

y3

,

Ž 1.

where s is the remote stress, Pc is the magma chamber pressure, r is the chamber radius and l is the radial distance away from the chamber wall. At the chamber surface Ž l s 0., Eq. Ž1. simplifies to: st s s y Ž D Pcr2., where D Pc is the chamber overpressure Žs Pc y s .. At the chamber surface, st becomes tensional when Pc ) 3 s . With a non-hydrostatic remote stress field st at the chamber margins can become tensional when Pc is less than the lithostatic overburden stress ŽSammis and Julian, 1987.. Previous analyses of magma chamber instability assumed that failure occurs by the formation of tensional fractures, which develop when chamber inflation has stretched the surroundings to the point where st has a tensional value greater than the tensile strength of the wall rocks ŽBlake, 1981; Sammis and Julian, 1987; Tait et al., 1989; Sartoris et al., 1990; Parfitt et al., 1993.. In our experiments, however, failure occurred by the propagation of a pre-existing liquid-filled fracture in the chamber wall. This can occur when st is still in compression. Consequently, dykes nucleate on pre-existing magma-filled cracks Žor other perturbations in the chamber surface. at much lower magma pressures than those required for tensional fracturing. The greater ease of this mechanism of failure is a consequence of the pressure exerted by magma on the crack walls and the stress concentration at the crack-tip. Pre-existing cracks in the wall rocks are to be expected considering the ubiquitous pervasive fracturing of the crust. Additional fractures due to the thermal influence of the magma are also likely ŽFurlong and Myers, 1985.. The propagation of pre-existing magma-filled cracks is therefore the dominant mechanism of dyke nucleation.

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We follow here the commonly adopted approach of linear elastic fracture mechanics; a pre-existing crack in a homogenous elastic medium becomes unstable and propagates when the crack-tip stress intensity factor, K, exceeds the fracture toughness of the medium, K c . A narrow 2-D crack in the walls of a magma chamber, of length l and filled with magma at a uniform pressure P, has a crack-tip stress intensity factor given by ŽLawn, 1993.: K s 1.12 Ž P y st . 'l .

Ž 2.

Note that while chamber overpressure is defined as Pc y s , the crack overpressure is P y st . Laboratory measurements of K c on various rock types provide values on the order of 1 MPa m1r2 ŽAtkinson, 1984.. Linear elastic fracture mechanics is not perfectly applicable to rocks at high confining pressures; consequently K c is not a simple material property and increases at high pressures ŽRubin, 1995b.. All else being equal, the longest crack has the highest K and so is the first to propagate. Eqs. Ž1. and Ž2. provide a model for a threshold of instability appropriate for a static crack at a uniform pressure equal to the chamber pressure. This corresponds to a situation in which the crack can be thought of as being able to respond instantaneously to changes in chamber pressure. However, our experiments illustrated that substantial times can be required for cracks to achieve pressure equilibrium with the chamber. The dynamics of this more realistic behaviour are now considered. In order to describe the flow of magma into the crack, a simple rectangular geometry is assumed ŽFig. 6.. The average flow velocity across the crack width, u, is approximated by the standard slot flow expression: us

w 2 Ž Pc y P . 3m l

Ž P y st . l m

,

where m is the elastic stiffness of the wall rocks. This expression is strictly appropriate for the minor axis of an elliptical crack at a uniform pressure. The application here assumes that taking w to be a representative width for the whole length of a rectangular crack is an acceptable approximation of the actual geometry, which in reality is a function of the along-crack pressure gradient. As magma flows into the crack from the chamber, the crack pressure rises. In response, the crack dilates to a new width; l remains constant Žuntil the crack-tip failure conditions are attained.. The volume of magma within the crack Ž V . increases according to: dV dt

Ž 3.

Ž 4.

s2

dw dt

lB,

Ž 5.

where t is time and B is a unit crack dimension perpendicular to l and w. The volume flux of magma into the crack can also be written in terms of: dV

,

where w is crack half-width, l is the crack length, Pc is the pressure in the magma chamber and at the crack entrance, P is the pressure at the crack-tip and m is magma viscosity. The width of a crack can be related to its length and pressure by ŽPollard and Segall, 1987. ws

Fig. 6. Diagram of the simplified characteristics of a magma-filled crack in a chamber wall.

dt

s u2 wB.

Ž 6.

Equating Eqs. Ž5. and Ž6. and combining with Eqs. Ž3. and Ž4. gives: dP s dt

3 Ž P y st . Ž Pc y P .

3m2m

.

Ž 7.

This is the differential equation governing the evolution of the crack pressure. The rate of crack pressurization is independent of crack length and lithostatic loading. There is a direct dependence on magma

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viscosity, for which there is a vast variability. Note that because the volume of the crack is negligible with respect to that of the chamber, Pc and hence st are unaffected by the flow of liquid into the crack. In order to integrate Eq. Ž7., Pc needs to be a known as function of time. An important simple case is that in which the liquid reservoir is suddenly pressurized, and then remains constant. In this case, Eq. Ž7. can be integrated with respect to time to yield: ts

3m2m

Ž Pc y st .

Ž Pc y st . 3

y2

1 y

Ž Po y st . 1 y

Ž Po y st .

/

2

/

2

ž

1

Ž P y st .

y Ž Pc y st .

ž

2

1

Ž P y st .

q ln Ž P y st . y ln Ž Po y st .

yln Ž Pc y P . q ln Ž Pc y Po . ,

Ž 8.

where Po is the crack pressure at t s 0, i.e., immediately after the chamber pressure has been increased to the value Pc . Eq. Ž8. defines the P–t evolution of the crack; the time delay until dyke propagation occurs is determined by when P satisfies the conditions for failure in Eq. Ž2.. 4.1. Comparison of experiments and theory The theory presented above can be used to calculate the pressure history of a crack in the margins of a chamber up to the point of failure and the onset of propagation. Eq. Ž8. will now be used to calculate the time delay before dyke propagation commences for comparison with the data from the second set of experiments in which chamber pressure was abruptly increased and then held constant. In order to apply the theory to the experimental data, the failure threshold Ži.e., the fracture toughness, K c . of the gelatine must be specified. We estimate that K c for the gelatine used in the experiments is approximately 45 Pa m1r2 . This is based on assuming that dyke nucleation occurred when the crack and chamber pressures were in equilibrium Ži.e., P s Pc . in the experiments involving the lowest viscosity liquid in the first set of experiments. The times calculated are

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only weakly sensitive to the potential uncertainties in K c . We assume that the crack and chamber pressures were in equilibrium before the abrupt increase in chamber pressure occurred. The change in st during the abrupt chamber pressure rise causes a corresponding depressurization of the crack. To account for this, we assume that the crack overpressure immediately after the increase in chamber pressure is equal to the crack overpressure before the increase, i.e., the liquid in the crack depressurizes by an amount equal to the change in st . The observed and predicted time delays before dyke nucleation for the second set of experiments are shown in Fig. 5. The calculations are for the specific conditions of each experiment, the kink in the theoretical curve is due to differences in the initial conditions for the highest viscosity liquid experiment. The theoretically predicted times are approximately an order of magnitude shorter than those observed in the experiments. One source of this discrepancy is the simplifications included in the theoretical analysis, but there are three other factors which can result in the experimental times being longer: Ž1. in the experiments, the increase in chamber pressure was not instantaneous, as flow into the chamber from the external reservoir took a finite time ŽSection 3.2.. Ž2. The values of the observed time delay are probably overestimates, due to the uncertainty in identifying the start of propagation of dykes which were initially very small and propagating very slowly. Ž3. The theory assumes that, prior to the sudden increase of Pc , the crack is in equilibrium with the chamber, whereas in the experiments this is extremely hard to ensure as it cannot be verified. A greater degree of disequilibrium is probable for the higher viscosity liquids. An initial amount of pressure disequilibrium would cause the theory to underestimate the time delay before propagation. All three of these difficulties conspire to cause the same sense of discrepancy between the experimental results and the predictions of the theory. Furthermore, none of these sources of disparity are relevant to the geologic application. Hence, given that the total discrepancy is only an order of magnitude, we consider that the theory can give approximately accurate estimates for the timescales that are associated with the process of dyke nucleation from crustal magma bodies.

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5. Dyke nucleation 5.1. Magma chamber pressurization and initial crack conditions In the following discussion, we concentrate on the general features of the behaviour described by Eqs. Ž7. and Ž8. rather than attempting to make quantitative predictions for specific examples. This is because application of the analysis to any specific geologic example requires knowledge of the rate of magma chamber pressurization and the initial crack conditions. There are currently only poor constraints on these variables in the geologic context which greatly reduces the interest of more precise calculations for specific examples at the present stage. Relatively rapid ground surface deformations have been detected above active magma chambers Že.g., Yang et al., 1992; Langbein et al., 1993; Thornber et al., 1996.. These are evidence of comparatively sudden and swift chamber pressurization and inflation. The most likely source of a relatively rapid increase in chamber pressure is a magma input from a deeper source; due to the increased stored mass of magma, and possibly also due to the exsolution of volatiles. Exsolution can be caused by: heating of the original magma by the fresh hot input, which lowers volatile solubility; the convective stirring associated with heating which may result in decompression-driven exsolution; and volatile transfer from the quenched input magma ŽSparks et al., 1977.. Magnitudes and rates of pressure increases during replenishment are poorly constrained and may well vary widely. They depend on the relative size of the magma input compared with the initial chamber, the magma flux during replenishment, and factors which influence input-driven exsolution Žinitial saturation state of the original magma and the temperature contrast with the input.. A sudden increase in chamber pressure may also occur without replenishment if convective overturning of the magma results in rapid decompression-driven exsolution ŽThornber et al., 1996.. In calculations presented in Section 5.2, which consider dyke growth following a sudden chamber pressurization, a range of chamber overpressures have been employed to illustrate the general features of dyke nucleation.

Slow changes in the chamber pressure conditions can result from magmatic evolution without replenishment. Crystallization increases the concentration of incompatible volatiles in the remaining melt, resulting in exsolution ŽBlake, 1984; Tait et al., 1989.. If magma is melting the surrounding crust, this may increase magmatic pressures as a result of the density changes associated with the phase transition. The rates of crystallization, exsolution and melting, and the associated pressure increase, are relatively gradual and dependent on the magmatic thermal evolution. This is determined by the balance between conductive heat loss to the surroundings ŽJaeger, 1968., heat lost in melting the chamber margins ŽHuppert and Sparks, 1988a,b; Kerr, 1994; McLeod et al., 1996; McLeod and Sparks, 1998., and the heat liberated by solidification. Once a dyke forms and the subsequent eruption relieves the chamber to zero overpressure, the dyke will close and solidify, and there will be another period of chamber overpressure increase until the conditions for failure are attained again. In the calculations of dyke growth during gradual chamber pressurization presented in Section 5.3, we consider a single linear chamber pressurization rate of 1000 Parday. We make no claim that this value is typical, our aim is solely to illustrate the general relationship between chamber and crack pressures during a gradual increase in chamber pressure. This rate is within plausible limits. For example, the magma supply rate of Kilauea is on the order 1 m3rs Že.g., Dvorak et al., 1983., assuming a total reservoir volume of 80 km3, this generates a pressurization rate of ; 10 000 Parday. Pressurization rates for more viscous magmas are likely to be slower, as some of the mechanisms of chamber pressure increase discussed above are viscosity dependent. The initial crack conditions depend on factors such as the previous magmatic history, the country rock type and structural setting; these cannot be easily specified in a simple or general way. All else being equal, the longest cracks that are available in the chamber margins are the least stable. As only the portion of a crack containing unsolidified magma is relevant to the analysis, lengths may depend on the temperature gradient in the wall rocks. The appropriate crack lengths are probably in the range of centimetres to metres. As a simple method of illustrating

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the basic behaviour of the model, we arbitrarily specify the initial crack pressure to be equal to the lithostatic load, i.e., Po s s and Po y st s D Pcr2 Žnote that as the crack is depressurized by the change in st associated with an increase in chamber pressure, a value of Po s s corresponds to a crack which was overpressurized relative to s before the increase in chamber pressure occurred.. The influence of different initial conditions will also be briefly examined. 5.2. Dyke growth following sudden chamber pressurization We consider the case where the chamber pressure is instantaneously increased at t s 0. As crack lengths are usually negligible compared to chamber dimensions, Ž r q l .rr is approximated as 1. The elastic stiffness of the crust is taken as 10 GPa ŽBieniawski, 1984.. The viscosities of basaltic and rhyolitic magma are assumed to be 80 and 10 7 Pa s, respectively. The solid curves of Fig. 7 shows the variation in crack overpressure following an increase in chamber overpressures to 0.5, 1, 2, 5 and 10 MPa. The crack pressure evolution is governed by the variation in the rate of magma flow into the crack. This is controlled by the opposing influences of the decreasing pressure contrast driving flow Ž Pc y P ., and the concurrent increase in crack width. Pressurisation rates are much slower for the smaller chamber overpressures, due to the relatively low pressure gradient driving flow and the narrow width of the cracks resulting from the small crack overpressures that are attained. The great sensitivity to crack width and hence to pressure leads to the strikingly non-linear nature of the curves in Fig. 7. Pressurisation rates for basaltic magmas are approximately five orders of magnitude quicker than for rhyolitic magmas, corresponding to the difference in viscosities. The variation in crack pressurization rate as a function of viscosity and chamber overpressure leads to corresponding differences in the time delay until the crack-tip fails and dyke propagation is initiated. Predictions of the time delay before the growth of basaltic dykes vary from, for example, 40 s Žfor Pc y s s 10 MPa and l s 1 cm. to 5 days Ž Pc y s s 0.5 MPa, l s 2 m.. For the same conditions, the delays for rhyolitic magma are 56 days and 1800

Fig. 7. Crack overpressure evolution following an instantaneous increase in chamber pressures at t s 0. Solid curves are for Po s s , and chamber overpressures Ž Pc y s . are as indicated. Dotted curves are for Po s Ž s q st .r2 and the dashed curves are for Po s Ž s q Pc .r2, with chamber overpressures of 1 MPa. Note the different scales on the horizontal axes. Ža. Basaltic magma. Žb. Rhyolitic magma.

years, respectively. Dyke nucleation may not occur, even when the crack pressure has equilibrated with the chamber, if the maximum crack pressure and crack length are insufficient to attain the failure criteria. The dotted curves on Fig. 7 show the crack pressure evolution due to a 1 MPa chamber over-

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pressure where the initial crack pressure is less than that of the lithostatic load Žin this example Po s Ž s q st .r2.. The dashed curves are for an initial crack pressure greater than the lithostatic load Žin this example Po s Ž s q Pc .r2.. Equilibration occurs more rapidly for the higher initial crack pressure. This is a result of a more rapid crack pressurization being promoted by the larger crack width, while the influence of the lower pressure gradient driving flow has a lesser impact. Variation in the choice of initial crack pressure has influenced the details of the predictions but not the general behaviour. 5.3. Dyke growth during gradual chamber pressurization Fig. 8 illustrates the general features of crack pressurization during a gradual increase in chamber

Fig. 8. Variation in the difference between the stress across the crack Ž st . and the chamber pressure Ž Pc . and crack pressures Ž P ., during gradual chamber pressurization at a rate of d Pc rd t s1000 Parday. Ža. Basaltic magma. Žb. Rhyolitic magma.

overpressure for basaltic and rhyolitic magmas. The initial conditions are that both the crack and chamber are pressurized to 1000 Pa in excess of the lithostatic load. The crack pressure evolution is then calculated iteratively for a chamber pressurization rate of 1000 Parday. The rise in crack pressure is initially slow, and then more rapid as the pressure contrast driving flow Ž Pc y P . increases. Eventually, the crack approaches near-equilibrium with the chamber. Large differences between crack and chamber pressures are sustained for relatively longer periods if the chamber pressurization rates are rapid andror the magma has a high viscosity. For the basaltic example, and assuming a crack length of 0.5 m, a dyke is not initiated until the crack and chamber are in near-equilibrium Žless than 100 Pa difference.. However, for the rhyolitic magma under the equivalent conditions, failure of the cracktip occurs when there is still a substantial disequilibrium, with a difference between chamber and crack pressures Ž Pc y P . of 2.34 MPa. This is a significant discrepancy compared to the 1.26 MPa crack overpressure Ž P y st . required to cause failure. The initial pressure contrast driving dyke propagation Ž Pc y st . in the rhyolitic case is almost three times that of the equivalent basaltic situation. The formation of the initial crack may involve non-elastic processes, such as wallrock melting and magma solidification at a partially molten chamber margin. For small cracks where non-elastic opening contributes significantly to the crack width, the theoretical analysis will underestimate crack pressurization rates. This may result in a smaller disequilibrium between chamber and crack pressures when dyke propagation commences. Although the exact degree of disequilibrium depends on the specific conditions, all else being equal there is a greater disequilibrium during the nucleation of silicic dykes due to their higher viscosities. This difference may be moderated by slower pressurization rates for chambers containing relatively silicic magma; although note that for stratified chambers different magma viscosities may apply to the processes of chamber pressurization and dyke nucleation. Some general consequences of a greater disequilibrium between crack and chamber pressures for more silicic magmas are longer repose times between eruptions, to allow the necessary higher cham-

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ber overpressures to develop, and the pressure contrasts driving dyke propagation and eruption are greater, presumably leading to more substantial eruptions. This is consistent with the observation that eruptions from silicic volcanoes are often less frequent and more voluminous than those from mafic centres.

6. Discussion and conclusions Probably the most significant and surprising conclusion is that dyke nucleation is a dynamic process, rather than being determined by a static failure criterion which is purely dependent on the elastic stress state in the crust surrounding a magma chamber. Magma viscosity plays a crucial role in determining the rate at which a crack in the chamber walls pressurizes, and so influences the timing of dyke nucleation and the chamber overpressures that are required for this to occur. In the light of the very large range in the viscosities of naturally occurring magmas, this has broad implications for assessments of volcanic hazards that are based on observations of ground deformation. If, for example, the inflation of a chamber as a result of a pressure increase Žperhaps due to replenishment. is detected by uplift of the ground surface, then subsequent predictions of dyke growth need to take account of the likely magma viscosity Ži.e., magma composition.. The delay before dyke nucleation following a sudden chamber pressure increase varies from typically hours to days for basaltic magmas, to many years for rhyolitic varieties. Where chambers are gradually pressurizing due to magmatic crystallization and evolution, the actual value of chamber overpressure at the point when the walls burst and a dyke forms depends on the magma viscosity. The relatively rapid rates at which basaltic cracks pressurize are likely to be quick compared to the rate of chamber pressure increase; consequently, crack and chamber pressures are probably relatively near to equilibrium when dykes form. However, the much higher viscosities of silicic magmas leads to significantly slower rates of crack pressurization, which probably results in a greater disequilibrium between crack and chamber pressures. Silicic magma

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chambers will continue inflating to some higher chamber overpressure before the crack pressure reaches the level required for failure. An additional process, which we have not quantitatively considered, but which could exacerbate the above effect is that of solidification. In the case of a slowly pressurizing crack containing silicic magma, solidification in the crack may significantly reduce the crack size. Higher overpressures are consequently required ŽEq. Ž2.. before the crack-tip fails. Rubin Ž1993b; 1995a. observed that, for equivalent driving pressures, rhyolitic dykes are more likely to freeze before they have progressed far from the source, whilst lower viscosity basaltic dykes can progress much further. This difference will be less if a larger pressure contrast drives the propagation of the rhyolitic dykes. The current results indicate that this is likely to be the case, allowing the potential for dykes to transport rhyolitic magma over substantial distances. Another factor promoting the extensive propagation of rhyolitic dykes is that there may be a significant increase in chamber pressure after dyke growth has begun, as a result of the relatively slow propagation rates ŽRubin, personal communication.. The chamber pressure variation is determined by two factors: the removal of magma into the dyke, and the continuing ‘background’ chamber pressurization due to other processes Že.g., crystallization, exsolution and melting of the surroundings.. When the dyke is relatively small, the low flux of magma into the dyke does not counteract the background pressurization within the chamber, and the chamber pressure continues to rise. Eventually, as the dyke lengthens and widens, the magma flux out of the chamber dominates and there is a net depressurization of the chamber. This can be illustrated by the following approximate analysis ŽRubin, personal communication.. Assuming a 2-D geometry, and neglecting fracture resistance, the magma flux out of the chamber can be approximated by D P 4 l 2 my3my1 , where D P is the pressure contrast determining both the propagation rate and the width of the dyke. The volume increase within the chamber as a function of the background chamber pressurization rate can be approximated by Žd Pcrdt .Ž2 r . 2 my1. The volume flux of magma out of the chamber into the dyke equals the rate of volume increase due to the

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background chamber pressurization, and so the chamber pressure is constant, when lr2 r ; Ž m2m Žd Pcrdt . D Py4 .1r2 . Assuming a background chamber pressurization rate of d Pcrdt s 1000 Par day and D P s 1 MPa, this condition is attained for a relatively short basaltic dyke when lr2 r ; 0.01, but for a much longer rhyolitic dyke of lr2 r ; 3.5. This indicates that, for rhyolitic magmas, there will be a more substantial increase in chamber pressure before the onset of depressurization Žnote that a prediction of the specific length a dyke grows to before depressurization must account for the variation in D P and l .. Solidification will reduce the dyke size and magma flux. This also promotes, for slowly propagating viscous dykes, the attainment of greater chamber overpressures before depressurization begins. The viscosity dependence of dyke nucleation promotes dykes to originate from the lower, more mafic sections of stratified chambers, where failure is either quicker or requires less overpressure. However, another aspect of chamber stratification, the magma density variation, and other factors which influence the stress distribution around the chamber Že.g., chamber morphology, size and depth. also have an effect on where dykes form ŽSartoris et al., 1990; Parfitt et al., 1993; McLeod, 1999.. When the stress field surrounding the chamber promotes failure of the roof, and there are strong viscosity contrasts within the chamber Že.g., a mafic magma chamber with a silicic roof layer produced by crustal melting., dykes may originate from just below the interface between the two magma layers; which may account for some eruptions of mingled magmas. If exsolution has been sufficient to produce a separate gas or foam layer at the chamber roof, this will facilitate fracturing by virtue of its low viscosity. The development of high overpressures for silicic magma chambers promotes extensive chamber inflation, which may lead to the chamber acquiring a more equi-dimensional form. In instances where very high chamber overpressures are attained, the stresses imposed on the surrounding crust may be sufficient to instigate alternative styles of brittle deformation; such as the growth of the chamber by slip along either new fractures or re-activated faults. This deformation may relieve chamber conditions sufficiently so that dykes do not form, or, eventual dyke growth may still occur if the minimum overpressures

required for continued fault-slip is greater than that needed for dyke nucleation. The theoretical analysis presented above provides constraints on the failure criteria of magma chambers. A long term aim is to use this understanding of the failure conditions to devise more quantitative prediction of when dyke growth occurs. This will probably require the development of an analysis of ground deformation and other geophysical data which can provide sufficiently detailed information on the variation in chamber pressure conditions for specific examples.

Acknowledgements PM was supported by a European Commission Marie Curie Fellowship. We thank Claude Jaupart and Thierry Menand for stimulating discussions. Allan Rubin and Paul Delaney provided interesting and constructive reviews.

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