Magmatic Processes: Physicochemical Principles© The Geochemical Society, Special Publication No.1, 1987

Editor B. O. Mysen

Physics of magma segregation processes

DoNALDL. TURCOTTEDepartment of Geological Sciences, Cornell University, Ithaca, New York 14853-1504, U.S.A.

Abstract-Partial melting occurs in the mantle due to pressure reduction or by volumetric heating.As a result magma is produced along grain boundaries at depths up to 100 km or more. Differentialbuoyancy drives this magma upwards by a porous flow mechanism. The small magma conduits ongrain boundaries may coalesce to form rivulets of magma in the asthenosphere. The reduction indensity due to the presence of magma may induce diapiric flows in the mantle. When the ascendingmagma reaches the base of the lithosphere it is likely to form pools of magma. Buoyancy drivenmagma fractures appear to be to the only mechanism by which this magma can penetrate throughthe lithosphere. How these fractures are initiated remains a subject of speculation.

INTRODUCTION mantle convection or above the core-mantleboundary for whole mantle convection (ALLEGREand TURCOTTE,1985). Pressure release meltingwould produce the observed volcanism. However,in many cases, the magma must penetrate the fullthickness of the cold lithosphere in order to producesurface volcanism.

We will consider three mechanisms for magmamigration: (I) porous flow, (2) diapirism and (3)fracture. Aspects of the magma migration problemhave been reviewed by SPERA(1980) and TUR-COTTE(1982).

PLATETECTONICSprovides a framework for un-derstanding magmatic activity on and in the Earth.However, many important aspects of magmaticprocesses are poorly understood. A large fractionof the Earth's volcanism is associated with mid-ocean ridges. Over a large fraction of the ridge sys-tem this is essentially a passive process. As the platesdiverge mantle rock must ascend by solid state creepto fill the gap. Averageupper mantle rock is sampledby this random, unsteady process. As the mantlerock ascends melting occurs due to the adiabaticdecompression. A substantial fraction of the re-sulting magma ascends to near surface magmachambers due to the differential buoyancy. This Adiabatic decompression will lead to partialmagma solidifies to form the oceanic crust. melting of the mantle on grain boundaries. Exper-

Volcanism is also associated with subduction imental studies (WAFF and BULAU, 1979, 1982;zones. The origin of this volcanism is still a subject COOPERand KOHLSTEDT,1984, 1986; fuJII et al.,of controversy. Friction on the slip zone between 1986) and theoretical calculations (VONBARGENthe descending lithosphere and overlying plate pro- and WAFF,1986) have shown that the melt formsvides some heat. However, as the rock solidus is an interconnected network of channels along tripleapproached, the viscosity drops to such low levels junctions between grains. Magma will migrate up-that direct melting is not expected (YUEN et al., wards along these channels due to differential1978). Offsets in the linear volcanic chains asso- buoyancy. This configuration suggeststhat a porousciated with subduction zones correspond with vari- flowmodel may be applicable for magma migrationations in the dip of the Benioffzones. This is strong in the region where partial melting is occurring.evidence that the magma is produced on or near Porous flow models for magma migration havethe boundary between the descending lithosphere been given by FRANK(1968), SLEEP(1974), TUR-and overlying plate. This magma then ascends ver- COTTEand AHERN(1978a,b), and AHERNand TUR-tically 150 ± 25 km to form the distinct volcanic COTTE(1979). Implicit in these models is the as-edifices observed on the surface. How the magma sumption that the matrix is free to collapse as thecollects, ascends and feeds the distinct centers is magma migrates upwards. The validity of this as-poorly understood. sumption has been questioned by MCKENZIE

Volcanism also occurs within plate interiors. A (1984). Ithas been shown by ScOTTand STEVENSONfraction of this volcanism may be associated with (1984) and by RICHTERand MCKENZIE(1984) thatascending plumes within the mantle. These plumes the coupled migration and compaction equationsmay be the result of instabilities of the thermal have unsteady solutions (solitons). However, it wasboundary layer at the base of the convecting upper shown approximately by AHERNand TuRCOTTEmantle, either at a depth of 670 km for layered (1979) and rigorously by RIBE (1985) that com-

69

POROUS FLOW

70

paction is not important in mantle melting by de-compression. The melting of the ascending mantletakes place sufficiently slowly that the crystallinematrix is free to collapse by solid state creep pro-cesses. It is appropriate to assume that the fluidpressure is equal to the lithostatic pressure. Anydifference between the hydrostatic (fluid) pressureand the lithostatic pressure will drive the buoyantfluid upwards. The conclusion is that the unsteadysolutions are not relevant to magma migration inthe mantle.

In order to quantify magma migration in a poroussolid matrix, Darcy's law gives (TuRCOTTEandSCHUBERT,1982, p. 414):

b2</>t:..pgt:..v=---247l'l1m

where t:..v is the magma velocity relative to the solidmatrix, b is the grain size, </> is the volume fractionof magma (porosity), t:..p is the density differencebetween the magma and the solid matrix, g is theacceleration of gravity, and 11mis the magma vis-cosity. Taking b = 2 mm, t:..p = 600 kg/m" and g= 10 m/s2, the magma migration velocity is givenas a function of the magma viscosity in Figure 1for several values of porosity.

1010"

10"

10'"

vm 10.7

mS

10.8

10-·

10.10I

'1m,Poise

10' 10' 10'

10'

10'

10'vm

10' ~~

10

10 10' 10'"7m,Pa s

FIG. 1.The solid linesgivethe magma migration velocityVm as a function of the magma viscosity '7rn for severalvalues of the volume fraction of magma (porosity) t/>. Thisresult from Equation (1) for the porous flowmodel assumesthat the grain size b = 2 mm and density difference D.p= 600 kg/m", The dashed lines give the magma migrationvelocity Vrn as a function of the magma viscosity '7rn forseveral values of the upward velocity of the mantle beneaththe melt zone VS' This result from Equations (I) and (2)assumes the melt fraction! = 0.25. The solid dot illustratesa particular example. Specifying '7rn = 10 Pa sand e, = 1ern/year determines the position of the dot which givesVrn = 10-7 mjs (300 cmjyear). The corresponding valueof the porosity is found from the intersecting solid line,for this example t/> = 0.003.

D. L. Turcotte

The upward flux of magma can be related to theupward flux of the component to be melted by

vm</>=vf3 s(2)

(1)

where Vm is the magma velocity, Vs is the upwardvelocity of the mantle prior to melting, andfis thefraction of the mantle that is melted. By usingEquation (1) to relate Vm and </> and takingf = 0.25,the appropriate values of Vm and </> are given in Fig-ure I for specified values of 11mand Vs. For example,with Vs = I ern/year and 11m= 10Pa s, we find that</> = 0.003 and Vm = 10-7 m/s (300 cm/yr) as in-dicated by the solid dot in Figure I.

Caution should be used in the application of theporous flowmodel. The theory assumes a uniformdistribution of small channels and in fact the smallchannels may coalesce to form larger channels asthe magma ascends in much the same way thatstreams coalesce to form rivers. There is some ev-idence that this occurs when ice melts.

DIAPIRISM

Diapirism has long been considered as a mechanismfor magma migration (MARSH,1978, 1982;MARSHand KANTHA,1978). Two types of diapirism mustbe considered. First, consider magma that has beensegregated from the mantle by the porous flowmechanism. The ascending magma is likely to poolat the base of the lithosphere. The question iswhether the buoyancy can drive the pooled magmathrough the lithosphere as a diapir. In order for thisto occur, the country rock must be displaced bysolid state creep so that the viscosity of the mediumthrough which the diapir is passing governs the ve-locity of ascent. A second question is whether theresidual partial melt fraction in the asthenospherecan induce diapirism.

In order to consider these problems we study theidealized problem of a low viscosity, low densityfluid sphere ascending through a viscous mediumdue to buoyancy. The velocity of ascent for thisStokes problem is given by (TURCOTTEand SCHU-BERT,1982,p. 267)

(3)

where, r, is the radius of the sphere and, 11.,is theviscosityof the medium. The ascent velocity is givenas a function of the viscosity in Figure 2 for severalvalues of the sphere radius and melt fraction andt:..p = 600 kg/m",Because the lithosphere certainly has a viscosity

of at least 1024Pa s, the migration velocity spheres

Magma segregation

'1s,Poise

10'" 10'8 10'0 1022

FIG. 2. The ascent velocity of a low viscosity, sphericaldiapir, Vd, as a function of the viscosity of the mediumthrough which it is rising, '7" for several values of the sphereradius, r, and porosity t/>. This result from Equation (3)assumes D.p = 600 kg/nr'.

with a radius less than 100 km is extremely small.The diapir will solidifybefore it can migrate througha cold lithosphere. If the path through the litho-sphere was heated so that the viscosity was lowered,then significant migration may occur. But what isthe mechanism for heating if magma cannot pen-etrate the lithosphere? The conclusion is that dia-pirism is not a viable mechanism for the migrationof magma through the lithosphere.

One possibility for this type of migration is thatheat transported from the magma or heating by vis-cous dissipation, or both, softens the mediumthrough which the diapir is passing. Studies of thisproblem (RIBE, 1983; EMERMANand TURCOTTE,1984; OCKENDONet al., 1985) show that thesemechanisms do not significantly help in the pene-tration of cold lithosphere.

Results given in Figure 2 can also be used to de-termine whether the melt fraction in the astheno-sphere can induce diapirism. Asthenosphere vis-cosities may be as low as 1018 Pa s although 1021

Pa s is probably a better estimate. With 11. = 1021

Pa s, </> = 0.01, and r = 100 km, we find Vd = I cm/yr. Under these conditions, diapirism due to partialmelt is relatively unimportant. However, with 11.= 1018 Pa s, the velocity may approach 103 cm/yr.Thus it is not possible to make a definitive conclu-sion on the relative role of diapirism within the as-thenosphere. To treat this problem satisfactorily,the mantle convection problem must be solved si-multaneously with the melting and porous flowmi-gration problems and this has not been done so far.

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MAGMA FRACTURE

A third mechanism for magma migration is fluidfracture. This is probably the only mechanism ca-pable of transporting magma through the coldlithosphere. The common occurrence of dikes isdirect observational evidence for the existence ofmagma fractures. The dynamics of pressure-drivenfluid fractures has been studied by SPENCEandTuRCOTTE(1985) and EMERMANet al. (1986).These authors showed that dike propagation is re-stricted by the viscosity of the magma and by thefracture resistance of the media, but in most geo-logical applications, the fracture resistance can beneglected.

The vertical transport of magma over large dis-tances (10-100 km) is almost certainly driven bythe differential buoyancy of the magma. One ofthemost spectacular examples of buoyancy drivenmagma fracture is a kimberlite eruption; the re-quired velocities are estimated to be 0.5-5 m/s(PASTERIS,1984). Buoyancy-driven fluid fracturesas a mechanism for magma migration have beenstudied by WEERTMAN(1971), ANDERSONandGREW(1977), ANDERSON(1979) and SECORandPOLLARD(1975). The dynamics of a buoyancydriven fluid fracture have been given by SPENCEetat. (1986). These authors found that the stress in-tensity factor (fracture resistance) plays an essentialrole in the solution.

The propagation of a buoyancy-driven fluidfracture is governed by the equations for the fluidflow in the crack; the equation governing the de-formation of the elastic medium, and the equationrelating the empirically derived stress intensity fac-tor to the tip curvature of the crack. Mathematicaldetails for the two-dimensional problem have beengiven by SPENCEet al. (1986). These authors finda steady-state solution for a propagating crack. Auniversal shape is obtained in terms of non-di-mensional variables. The non-dimensional crackwidth 2H is given as a function of the non-dimen-sional distance from the crack tip in Figure 3. Thenon-dimensional variables are related to actualvariables by

H='!!:_, X=([I-V]t:..pg)1/2Xhoo hoo µ.

(4)

where 2hoo is the width of the tail, µ. the shear mod-ulus, v Poisson's ratio, t:..pthe density difference andg the acceleration of gravity. One of the importantresults of the analysis is the requirement that thecrack must have an infinitely long constant widthtail. The flow in the tail is a balance between thebuoyancy driving force and the viscous resisting

72

-4 -3 -2 -I o

4

5

~------~~6~------~FIG.3.Shape ofan upward propagating magma fracture.

The non-dimensional half-width H is given as a functionof the non-dimensional distance from the crack tip X.The non-dimensional variables are defined in Equa-tion (4).

force. Thus the velocity of propagation of the crack,c, is related to the half-width of the tail hoo by

gh~t:..pc=--311m

for laminar flow in the crack and

for turbulent flow.For the tail of the crack, the non-dimensional

half-width is Hoo = 1. The maximum crack half-width is H = 1.8975 and this occurs at X = 1.152.A steady-state solution to this problem is foundonly for a singlevalue of the non-dimensional stressintensity factor, A, defined by

x =~1~ ((1 - V))3/4 x;21/2 µhoo (gt:..p)I/4 = 1.3078

whereK; is the critical stress intensity factor. Solvingfor the half-width of the tail gives

K~/3(1- v)hoo = 0.440 µ.(gt:..p)I/3 .

D. L. Turcotte

4 When the critical stress intensity factor is specified,the half-width of the crack is obtained from Equa-tion (8). The propagation velocity is then obtainedfrom Equation (5) for laminar flow and Equation(6) for turbulent flow.

The critical stress intensity factor, Xc, is a measureof the fracture toughness of the material. Crackscan propagate at low velocities (_10-4 m/s) by themechanism of stress corrosion; in this range K < Xc.However, when K = Xc, crack propagation becomescatastrophic and the propagation velocity may rap-idly accelerate to a large fraction of the speed ofsound. The values of K; for a variety ofrocks havebeen obtained in the laboratory. This work has beenreviewed by ATKINSON (1984). Measured values forgranite range from K; = 0.6 to 2.2 MN/m3/2 andfrom K; = 0.8 to 3.3 MN/m3/2 for basalts.A measured value for dunite isK, = 3.7 MN/m3/2.These values were obtained at atmospheric pressure.The influence of pressure on the stress intensity fac-tor is difficult to predict. SCHMIDT and HUDDLE(1977) found a factor of four increase at a pressureof62 MPa for Indiana limestone. Thus, it is difficultto specify values of the critical stress intensity factorfor regions of the crust and mantle where buoyancy-driven magma fracture is occurring.

By taking µ. = 2 X 1010Pa, V = 0.25, t:..p = 300kg/m? and g = 10 m/s2 the half-width of the tailfrom Equation (8) is given as a function of the crit-ical stress intensity factor in Figure 4. Propagationvelocities from Equations (5) or (6) are given as afunction of the critical stress intensity factor in Fig-ure 5. Results are given for magma viscosities 11m= 0.1, 3,100 Pa s. We use the laminar result equa-tion (5) for Reynolds numbers Re = PmChoo/l1m lessthan 103 and the turbulent result Equation (6) forReynolds numbers greater than 103. The corre-

(5)

(6)

(7)

10 102

KcMNm -3/2

(8)FIG.4. The half width of the crack tail hoc as a function

of the critical stress intensity factor K; from Equation (8)by assuming µ = 2 X 1010Pa, v = 0.25, D.p = 300 kg/rrr',

Magma segregation

cill5

10-8L.. j__ ...L... --'

I 10 102 103

KcMNm-3/2

FIG. 5. Crack propagation velocity, c, as a function ofthe critical stress intensity factor, K, for several values ofthe magma viscosity, '7rn. Laminar flow from Equation (5)and turbulent flow from Equation (6).

sponding values of the volumetric flow rate V= 2choo per unit crack length are given in Figure 6.

CARMICHAELet al. (1977) have shown that theentrainment of xenoliths in basaltic flows impliesvelocities, c, of at least c = 0.5 m/s. A variety ofstudies indicate that basaltic magma migrates up-ward at velocities in the range c = 0.5 to 5 m/s. Atypical viscosity for a basaltic magma is 11m = 0.1Pa s. From Figures 4 and 5 the corresponding rangeof the stress intensity factor is K; = 20 to 100 MN/m3/2 and the range of tail half-widths is b-. = 5to 50 mm. From Figure 6 the corresponding rangeof flow rates per unit length is V = 0.0025 to0.25 m2/s.

Studies of the Kilauea Iki eruption on an islandof Hawaii during 1959-1960 give flow rates of 50to 150 m3/s (WILLIAMS and McBIRNEY, 1979, pp.232-233). Taking a flow rate of 100 m3/s, the rangeof flow rates per unit length given above, V = 0.0025to 0.25 m2/s, correspond to crack lengths between400 m and 4 km. These appear to be reasonablelengths for the crack feeding Kilauea Iki. Althoughthe above comparison appears reasonable, there isconflicting evidence whether the flow of magma atthe surface represents the flow through the litho-sphere. The role of shallow magma chambers instoring magma is poorly understood.

As a specific example of a magma fracture thattransports magma through the lithosphere we take:µ. = 2 X 1010Pa, v = 0.25, t:..p = 600 kg/m", g = 10m/s2, 11m = 0.1 Pa s, and K; = 50 MN/m3/2. Wefind that hoo = 0.0167 m and c = 2.86 m/s. Toconvert the shape given in Figure 3 into actual dis-tances, we require h = 0.0167H m and x = 272X

73

m. If the length of the crack is 1 km the volumeflux through the lithosphere is 95.5 m3/s.

CONCLUSIONS

Pressure release melting produces magma ongrain boundaries. Magma on triple junctions be-tween grains produce an interconnected networkof channels. Under mantle conditions the magmadrains rapidly upwards due to differential buoyancyand the residual solid matrix collapses due to solidstate creep. Magma that is produced over a depthrange of approximately 50 km mixes during thevertical ascent to produce the magma reaching thebase of the lithosphere.

At mid-ocean ridges there is no lithosphere andthe ascending magma can form directly the oceaniccrust. However, off the axis at ocean trenches andat intraplate volcanic centers, the magma mustpenetrate through the lithosphere. Magma fractureis a mechanism for the rapid transport of magmathrough the lithosphere. A theory for steady statemagma fracture through the lithosphere is given. Atypical fracture width is 2 em and fracture velocityis 5 m/s. Under these conditions relatively littlemagma is lost by solidification during ascent. Theexistence of dikes is evidence that magma fractureis a pervasive mechanism. The extensive systemsof dikes in deeply eroded terranes is evidence thatdikes are the dominant mechanism for the ascentof magma through the crust. Kimberlite eruptionsare direct evidences that magma fracture can trans-port magma through the lithosphere.

How magma fractures initiate remains a matterof speculation. It may be possible to form magmafractures as magmas perculating in small channels

10-2

V, 10-4

m'5 10-6

10-12I 10 102

Kc,MNm -3/2 103

FIG.6 Volumetric flow rate per unit crack length, V, asa function of the critical stress intensity factor, K, forseveral values of the magma viscosity, '7m.

74

coalesce to form larger channels as suggested byFOWLER (1985). Alternatively, magma may pool atthe base of the lithosphere. When the pool reachesa critical size the buoyancy forces may initiate amagma fracture that subsequently drains the pool.

Acknowledgements-The author would like to acknowl-edge the many contributions of D. A. Spence to our un-derstanding of the magma fracture problem. This researchwas supported by the Division of Earth Sciences, NationalScience Foundation under grant EAR-8518019. This iscontribution 830 of the Department of Geological Sciences,Cornell University.

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