carbon dioxide in igneous petrogenesis: ii. fluid dynamics of mantle metasomatism

Contrib Mineral Petrol (1981) 77:56-65 Contributions to Mineralogy and Petrology �9 Springer-Verlag 1981

Carbon Dioxide in Igneous Petrogenesis: II. Fluid Dynamics of Mantle Metasomatism

Frank J. Spera

Department of Geological and Geophysical Sciences, Princeton University, Princeton, NJ 08544, USA

Abstract. Petrographic, fluid inclusion, geochemical and isotopic evidence from xenoliths in alkali basalts suggests that low-viscosity fluids rich in O- -H- -C , dissolved silicates and especially the incompatible elements may ascend, decompress and precipitate crystalline phases and/or induce partial fusion in the upper mantle. Such mantle metasomatic fluids (MMF) may be important in generating isotopic heterogeneity and in transporting and focusing mantle heat. In order to model the movement of MMF, the ordinary differential equations governing the variation of P, T, ascent velocity and fluid density of a compressible, viscous, single-phase (H20 or CO2) non-reacting fluid ascending through a vertical crack of constant width have been solved. A large number of numerical simulations were carried out in which the significant factors affecting flow behavior (thermodynamic and transport fluid properties, roughness and width of cracks, geothermal gradient, initial conditions, etc.) were systematically varied. The calculations show that: (1) MMF tends to move at uniform rates following a short period of rapid initial acceleration, (2) MMF ascends nearly isothermally, (3) MMF acts as an ef- ficient heat transfer agent; numerical experiments show that transport of heat into regions undergoing metasomatism can lead to partial fusion. The heat transported by movement of MMF averaged over the age of the Earth is sufficient to generate about 0.1 km 3 of basaltic magma per year, which is approximately equal to the production rate of alkaline magma. If an intense period of mantle degassing occured early in the history of the Earth, the transport of heat and mass (K, U, Rb, LREE) by migrating fluids might have been important.

Introduction

In the first of this series of publications (Spera and Bergman 1980) the thermodynamic properties of carbon dioxide in silicate melts were retrieved from experimentally determined solubility data for natural and other silicate melts. Specifically it was shown that

(1) The systems COz-tholeiite, CQ-andesite and CO 2- melilitite may be treated as ideal pseudobinary systems in which the simple activity - composition relationship ac~o2 = X~o 2 holds. The observation that mixing in these systems is ideal needs to be qualified. Indeed, no experimental evidence has been offered to show that COe component mixes ideally with each of the other oxide components in a silicate melt.

However, if the mole ratios of the other components remain constant (i.e., only Xco2/Xi varies), then the solubility data are consistent with ideal mixing in the pseudobinary melt-CO 2 system. The assumption of ideal mixing greatly facilitates the calculation of solubilities of CO 2 in melts at high temperatures and pressures.

(2) The partial molal volume of CO 2 in the melt (~co2) is surprisingly independent of Pt and T and is approximately 30 cm3mole -1 at high temperatures and pressures.

(3) ACe of the reaction CO2(rnelt)~-~.CO2(vapor) is nearly zero over the Pt T range relevant to magma genesis and transport.

(4) The solubility of CO 2 in silicate melts in highest in melts with low asio~ and high alkali activities; thermochemical predictions based on simplified polymerization (polycon- densation) reactions between CO 2 and silicate melt species are in good agreement with experimental data.

The calculated increase of CO 2 solubility at high pressure in melts of high and low alkali and silica activity, respectively, together with the long-standing petrological association of abundant volatiles in alkaline magma, suggest the fluids play an important role in the generation and ascent of these magmas. Indeed it has been inferred that some magmas (e.g., kimberlites) ascend quite rapidly from great depths. This in- ference is based on the capacity of kimberlites to transport large xenoliths from great depths. Simple calculations based on rates of graphitization of diamond (Anderson 1979) and settling rates of ultramafic, eclogitic or granulitic xenoliths in kimberlites (Spera 1980; McGetchin and Nikhanj 1973; Shaw, 1965) indicate characteristic ascent rates around 101 to 102m s -1 (see also Goetze 1975). These high ascent velocities cannot be attributed solely to buoyancy forces. Instead, it is suggested that a fracture mechanism involving brittle extensional failure (Shaw 1980) under conditions of low effective stress accompanied by viscous flow of a rheologically complex multiphase compressible magmatic suspension is most consistent with geological and petrological observations (Cloos 1941; Dawson 1980; Shoemaker 1956; McGetchin et al. 1968). Analysis of the equations of mass and momentum conservation (see below) indicates that in order for magma to be accelerated, and high flow rates maintained, the magmatic suspension must behave as a low-viscosity, compressible fluid. Homogeneous melts or crystal-melt suspensions are notable for being relatively incompressible. However, if volatile components initially dissolved in the melt phase exsolve, then the mean density of the mixture (melt+crystals+vapor) can drop quite dramatically and magma will accelerate. The dynamical effects of volatile exsolution and expansion are not

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necessarily limited to kimberlitic magma. Because the solubilities of H20 and CO 2 are so small at low pressure, it is expected that essentially any batch of magma will experience a period of rapid acceleration during the last part of ascent. The precise depth at which the transition from incompressible (crystals+melts) to compressible (crystals+melt+vapor) flow occurs depends on the total volatile content of the melt and the solubility expressions for the important volatile components. The latter expressions are strongly dependent on Pt, T, fo~, asio~, aNa2o, aK~o and other intensive thermodynamic parameters. The dynamical history of ascending multiphase magmatic suspensions is also very dependent upon the relative proportions of crystals, melt and vapor in the magma and may be highly variable. Kimberlites, for example, may consist of roughly equal portions of each constituent, whereas basaltic magma (e.g., MORB) is presumably melt-dominated at least until the last several hundred meters of ascent. Be- cause vapor viscosities are so small compared to melts or melt-crystal suspensions, one might expect a priori rather different transport behavior of a vapor versus melt-dominated system (e.g., kimberlite vs. basalt).

In a forth coming publication we consider the quantitative aspects of magma transport by exsolution of volatiles. Specifi- cally, we use H 2 0 - - C O 2 solubility data in conjunction with a fluid dynamic model of magma ascent to predict magma pressure, temperature, flow rate, mean density and the volume fraction and composition of low-density vapor phase as a function of depth, given an appropriate set of initial conditions. Calculations suggest that the almost universal limi- tation of mantle-derived xenoliths to alkaline magma is related to rapid sub-moho velocities. High ascent rates come about because of (1) relatively large concentrations of in- soluble gases (e.g., CO2, N2) in the melt and (2) tectonic (stress) conditions which favor extensional failure in the lithosphere. Small deviatoric effective stresses are sufficient to propel magma upwards at rates on the order of meters per second. These matters are discussed in a later publication. They are briefly mentioned here because the "movement history" of volatile-charged magma is closely associated with the main topic of this paper: an exploration the fluid dynamics, thermodynamic and transport properties and thermal history of low viscosity (r/~10 -3 poise) volatile-rich fluids in the upper mantle that bear on the problem of mantle metasomatism (Bailey 1970, 1980; Lloyd and Bailey 1975).

In the past decade considerable petrographic, isotopic and geochemical evidence has accumulated supporting the con- cept that many parts of the mantle have experienced a complex history of partial melting, intrusion, crystallization, re- crystallization, deformation and alkali metasomatism. Al- though the geochronological puzzle is difficult to piece together because of inherent sampling biases, chronologies based on textural and isotope data have been constructed for a number of xenolith suites. Well-documented case histories (Francis 1976; Best 1974; Witshire et al. 1980; Reid et al. 1975; Wilshire and Shervais 1975; Lloyd and Bailey 1975; Boyd and Nixon 1975; Erlank 1976; Harte et al. 1975; Sun and Hanson 1975; Frey and Green 1974; Menzies and Murthy 1980, Bergman 1981) reveal that the mantle is isoto- pically and chemically heterogeneous. Mineralogical and isotopic heterogeneities in the mantle are related to the complex igneous, metamorphic and metasomatic events that have af- fected the particular volume of sampled mantle. Of special significance to this paper is the well-documented process of mantle metasomatism in which ascending mantle metaso-

matic fluid (MMF) rich in Fe 3 +, Ti, K, Nb, REE, C, H, C1, F, O and other LIL elements tends to equilibrate with peridotitic mantle (Boettcher et al. 1979, 1980). A manifestation of chemical potential or thermal equilibration of mantle metasomatic fluid (MMF) with peridotitic mantle is the growth of metasomatic phases and/or the inducement of partial fusion. A detailed treatment of infiltration/diffusion metasomatism along the lines given by Korzhinskii (1965, 1970), Thompson (1959) and other students of metasomatism (e.g., see Fletcher and Hoffman 1974, Hoffman 1972; Helgeson 1971) is not possible until: (1) the composition, transport, thermodynamic and electrostatic properties of MMF are established and (2) the fluid dynamics of MMF ascent is better understood. One of the goals of this paper is to summarize briefly the properties of H 2 0 - - C O 2 rich fluids at upper mantle conditions of pressure and temperature that are relevant to metasomatism. A second is to propose a simple one- dimensional compressible fluid flow model for the decompression and ascent of MMF. The dynamic model explicitly accounts for the loss of fluid head due to frictional effects such as throttling, skin friction and other irreversible viscous flow processes. Heat-transfer between MMF and the surrounding lithosphere is also accounted for by exploiting well- established correlations between heat transfer coefficients and relevant flow parameters.

These considerations are a necessary first step towards a more complete theory of the physical chemistry and fluid mechanics of mantle metasomatic processes.

Composition and Solution Properties of MMF

Knowledge concerning the composition of MMF comes prin- cipally from the following sources:

1) Composition of phases formed by reaction of MMF with peridotitic mantle (references previously cited);

2) Composition of primary (high density) fluid inclusions found in tectonite, cumulate and intriguing 'composite' xenoliths and in megacrysts and phenocrysts found in alkali basalts (Murck et al. 1978; Roedder 1965; Sobolev et al. 1972);

3) Abundances of volatile elements in carbonaceous chrondrites (Mason 1969);

4) Composition of volatiles degassed from the interiors of the terrestrial planets (Pollack and Yung 1980);

5) Experimental studies of solute concentrations in H20 and H 2 0 + C O 2 fluid in equilibrium with silicate melt and crystalline phases at high P and T (Eggler 1973; Eggler and Rosenhauer 1978; Holloway 1971; Ryabchikov and Boettcher 1980);

6) Composition of volcanic gases and volatile contents of glassy volcanic rocks (Anderson 1975; Moore 1970).

Although none of these lines of evidence is firmly con- clusive, taken together they strongly suggest that the metasomatic fluid substrate is rich in O, H, C, S, F and C1 and must have an appreciable solute content.

The distribution of species in the system C O - - H - - S is dependent on bulk composition, T, P~, fo2 and values of other intensive variables that may be fixed by crystal-fluid phase equilibria (e.g., equilibrium co-existence of dolomite, orthopyroxene, olivine and clinopyroxene defines a unique %02 at a given P~, T). Calculation of fluid species equilibria in the system C - - O - - H - - S indicates that the species H20, CO2, H2, H2S , and CH 4 are most abundant (Gerlach and Nordlie

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1975; Anderson 1975). It is noted that the precise distribution of volatile species will dpend on the nature of the crystalline phases in equilibrium with the fluid. In the fluid dynamic calculations presented below, we assume that H20 and CO 2 make up the bulk of the fluid. However, the oxidation state of C is intimately related to the spatial variation of fo2, T, P~ and the solubility of C in silicate phases throughout the upper mantle, about which little is known.

The concentration and speciation of dissolved components in MMF is unknown. If the composition of phases that precipitate by sub-solidus reaction of M M F with mantle or crystallize from melts produced by anatexis of metasomati- cally - altered peridotites are reflective, then M M F must be enriched in Ti, Fe 3+, Nb, P, LREE, U, Th, F, C1, Na § K +, Cs § some alkaline earths and other LIL elements. It seems reasonable to suppose that solute abundances depend on P~, T, the precise composition of the substrate and the degree of MMF-peridotite equilibration.

Limited experimental data (Eggler 1973, 1975; Holloway 1971; Boettcher and Wyllie 1969; Morey 1957; Burnham 1967; Shettel 1973) suggest that an2o/aco 2 is a critical factor in determining the solubility of crystalline oxide, sulphide and silicate phases in MMF. Addition of small amounts of CO 2 to an H20-ric h fluid at t~500~ and P t ~ l k b dramatically increases the solubility of calcite and hematite whereas the concentration of dissolved silica decreases in the system C a O - - F e O - - S i O 2 - - H 2 0 - - C O 2 according to Morey. Shettel (1973) showed a similar negative correlation between silica solubility and acoJait2o in binary (CO2--H20) fluids. H20- pure vapor in equilibrium with diopside (Pt ~ 20 kb, t ~ 1,300 ~ C) contains about 12 wt. % solute (Eggler 1975) whereas pure CO 2 vapor contains < 1 wt. % solute at the same conditions (Eggler and Rosenhauer 1978). In the system KMg3A1S3010(OH)2-H20-CO2, Eggler found solute contents of 1 to 3wt.% at 25kb and 1,200~ for aco ~ >an~ o. On this basis he speculated that fluid in equilibrium with phlogopite could dissolve and transport alkalies and perhaps other LIL lelments. Recent experiments by Ryabchikov and Boettcher (1980) indicate a total solute content of about 15wt.% at 1,050~ and 20kb for H20-rich fluid in equilibrium with phlogopite, forsterite, Mg-rich spinel and orthopyroxene. They also found solute contents to depend rather markedly on P~, other variables (e.g., composition, temperature, etc.) remaining constant. Nakamura and Kushiro (1974) found that aqueous fluid in equilibrium with forsterite and orthopyroxene at 15kb and 1,300~ contains around 20wt.% SiO 2. The work of Holloway (1971) on the system basa l t -H20--CO 2 similarly showed a positive correlation between the vapor solute content and P~. In his experiments, elements enriched in the fluid phase relative to the basalt include A1, Mg, Na, K and Sr whereas Si, Ca and Fe 2+ show the reverse trend. Small amounts of certain ha- logen elements (e.g., F, C1) probably enhance mineral solubilities greatly (Burnham 1967).

We conclude by noting that H20-rich M M F probably contains significant amounts of dissolved solids especially at high pressure. The precise speciation depends on Pt, T, bulk composition, and electrostatic properties of the fluid (Hel- geson et al. 1974). Experimental data of Modreski and Boett- cher (1973), Boettcher and Wyllie (1969), and Ryabchikov and Boettcher (1980) suggest that complete miscibility between solute-bearing vapor (H20 rich) and water-saturated melt may exist at high pressures. If this is so, then solute contents of MMF at high Pt and T may be very large. Indeed,

a theoretical study by Spera (1974) predicts complete miscibility in the system Harding Pegmatite - H20 at 700~ for Pt>26kilobars. Isothermal P - -X saturation curves for coexisting melt and vapor phases are decidely asymmetric; for Harding Pegmatite the critical composition (Xc) is calculated to be X C (H20)=0.8 , which implies a solute content of about 50wt.% in the supercritical fluid. The point is that M M F can contain significant amounts of dissolved solids and that upon decompression it will tend to precipitate crystalline phases (e.g., phlogopite, apatite, perovskite, amphiboles) that are enriched in the incompatible and volatile elements. If mantle metasomatism is widespread, isotopic and trace element heterogeneity on a variety of spatial scales would be expected.

Migration of MMF : Thermodynamic and Fluid Mechanical Constaints

The purpose of this and the following section is to summarize information gathered from a large number of numerical experiments that model the expansion and vertical ascent of M M F within the lithosphere. The fluid dynamic model is a one-dimensional, steady-state approximation to the full set of equations expressing conservation of mass, momentum and energy for a single-phase fluid.

In the calculations, the metasomatic fluid has been taken to be either pure H20 or pure CO2; these are the only geologically important fluids for which reasonably well-constrained equations of state at the physical conditions of in- terest are available (Burnham et al. 1969; Helgeson et al. 1974; Spera et al. 1980; Holloway 1977; Shmonov et al. 1974). Since M M F may contain appreciable dissolved solids, calculations performed assuming pure CO 2 or H20 are obviously approximate. The effect of dissolving small amounts of solids in the fluid is not expected to greatly alter fluid properties.

In the calculations presented here, the fluid is assumed to be non-reacting. This must generally be a poor assumption. If it were not, there would be no petrographic evidence for mantle metasomatism. An obvious effect of chemical reaction between M M F and peridotitic host is the loss of fluid heat. High activities of certain components in the fluid phase presumably cause crystalline phases to precipriate as the fluid undergoes decompression and partial equilibration with peridotite. The eventual effect of these reactions on MMF will be to stop its upward migration Indeed if the ambient PT conditions are such that H20 and CO2-bearing crystalline phases are stable (e.g., amphibole, phlogopite, carbonates) one expects the growth of metasomatic volatile-bearing phases, inducement of partial fusion or some combination of both to 'consume' MMF. However, if M M F is rich in CO2, there may not be a tendency for 'consumption' of fluid as it ascends. For example, in sub-oceanic lithosphere at a relatively short distance from a spreading center, high mantle temperatures may preclude the stability of carbonate phases. If oxygen fugacities are high enough to stabilize CO 2 relative to CH4, C and other possible reduced compounds, then the existence of CO2-rich fluids is possible. On the other hand, CO 2 is not stable with respect to carbonates in cold, oxidized sub-continental lithosphere; one expects crystalline carbonates to be present, in an amount that depends on the local abundance of Carbon. CO2-rich fluids could be generated if temperatures were made to rise above the dolomite stability

curve defined by the reaction

enstatite dolomite diopside olivine vapor

4MgSiO 3 +CaMg(CO3) 2 ~ CaMgSi206+ 2MgzSiO4+CO 2 orthopyroxene carbonate clinopyroxene forsterite

Straightforward thermochemical calculations using data from Helgeson et al. (1979), and Huebner (1971) shows that the reaction

4 MgSiOa + CaMg(CO3)2~CaMgSi206 + 2 Mg2SiO 4 + 2 C + 2 0 2

has a large negative A G(A G~-60kcal ) for fo 'S along the iron-wustite (IW) buffer at t=1,250~ and P~=20kbar. In order for dolomite to be stable, fo2'S need be around QFM. If the intrinsic oxygen fugacity measurements of Arculus and Delano (1980) on mantle-derived spinels from peridotite are applicable, then carbonates would be unstable with respect to C (diamond), CH 4 and perhaps other species stabilized by strongly reducing conditions (see Perchuk 1976). Although these considerations are not explicitly considered in the fluid dynamic model presented below, they obviously influence to a great extent the composition and mobility of MMF.

Simultaneous solution of the three conservation equations and the equation of state for HzO or CO 2 gives the pressure, temperature, velocity and density of rising fluid as a function of height above the reservoir. Transport properties (e.g. thermal conductivity, dynamic viscosity) of the fluid are calculated from corresponding state correlations (Appendix III). The computation of a flow trajectory begins by assuming a set of initial conditions. It is assumed that M M F leaves a pressurized reservoir at P~, T~, and an initial vertical velocity ui. The fluid can be thought of as propagating and flowing up along a cylindrical crack; the details of the brittle failure phenomena associated with growth of a crack are not explicitly considered here. This question has been discussed elsewhere (Shaw 1980; Anderson 1979). It is reasonable to expect that their analysis is applicable to the migration of MMF.

Comprehensive transport calculations for a planar infinite crack have not been carried out; however, simple calculations suggest that the effect of fracture configuration (i.e., a planar versus cylindrical conduit) is most important only for very non-adiabatic flows. The precise history of ascent is determined by numerical solution of the equations governing flow. These solutions are summarized in a later section after a brief discussion of the initial conditions.

Necessary Conditions for MMF Migration

A fundamental constraint for the upward movement of MMF is that the fluid pressure (PI) be approximately equal to the mean normal stress at any depth. To a good approximation the mean normal stress may be identified with lithostatic pressure (Pz). If PI~Pz, rapid upward ascent becomes a mechanical impossibility. Pressurized M M F is mobile because of the ease with which fluid filled cracks may nucleate and rapidly propagate upwards. The tensile strength of peridotite rocks is on the order of several hundred bars and if deviatoric stresses are of this order, cracks can form and rapidly grow (Anderson and Grew 1977; Spera 1980). Low viscosity of M M F (Appendix I) insures that a continuous supply of fluid can be supplied to the tip of a rapidly (cm/s to m/s) propagating crack.

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The initial velocity of fluid out of a pressurized reservoir may be estimated from laboratory experiments on the rheo- logy of rocks under low effective stress conditions. If Py-P~ equal to 200 bars can be maintained over a kilometer vertical height, the relation

A PD 2 ui = 3 2 ~ - (1)

yields M M F initial velocities around 10m s -1, assuming the cracks to be about a c m in diameter. A simple calculation based on the stress drop and fault displacement suggests that failure of the reservior walls in this case generates an earthquake of approximate surface wave magnitude M s =2.3. (Die- terich 1974).

Steady-State Fluid Dynamic Model

The one-dimensional form of the differential equations governing the vertical variation of u, P, T and p of a compressible, viscous, single-phase non-reacting fluid that ascends in a vertical crack of constant diameter D may be written

du M dp dZ - p2 dZ (2)

dP I MdU 2C~.;VI 2 1 - - p g (3)

dZ dZ D p

dT e T - l d P s u du 4h g ( T - Tw) - - - (4)

dZ pCp dZ Ce dZ CpMD Cp

d o f dPr dT} d z - p lfl ~ - c ~ dZ_ (5)

These equations are derived in numerous textbooks (Shapiro 1953; Kay 1968). Equations (2)-(4) express conservation of mass, momentum and energy for fluid rising in a pipe, while Eq. (5) is a differential form of the equation of state of the fluid. Note that these equations are coupled to one another; solution of this set of ordinary differential equations is best accomplished by numerical (Runge-Kutta) methods. The exe- cution of a numerical experiment requires specification of the P~- T dependence of the thermodynamic (p, c~, fl, Cp) and transport (k, t/) properties of the fluid in addition to other relevant parameters (CI, h, Tw(Z)). The values adopted here are given in Appendix III. In the sections which imme- diately follow, remarks based on a scaling analysis of Eqs. (2) to (5) are presented. Following this, the implications of the numerical simulations to petrogenesis and heat transfer within the mantle are discussed.

The continuity equation indicates that the local speed of ascending fluid at any depth depends on the initial fluid mass flux (m=piui) from the reservoir and the local density of the fluid. The relation amongst these variables is given by

6u=-~ fi p (6)

where cSu and c~p represent the changes in velocity and fluid density respectively. Note that although ~/ is fixed by the initial conditions, p is a variable and consequently fluid accelerations will be large when p and dp/dZ change rapidly. Both HzO and CO 2 behave essentially as incompressible fluids at mantle PT conditions and so except for fluid accelerations associated with the failure of the reservoir (i.e., very

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early in the expansion process), M M F should tend to move at roughly a constant rate as it undergoes decompression.

Limits may be placed on the variation of fluid pressure (P~) as a function of height above the reservoir. There are two limiting cases:

(1) Fluid pressure is equal to mean normal stress (approximately equal to lithostatic pressure) at all depths; in this case the crack must propagate at a slow and constant rate (quasi- static crack propagation) and must be small in length compared to the overall fluid migration distance.

(2) Fluid pressure is governed by inertial, frictional and gravitational forces. The fluid is imagined to fill a long crack that remains connected to the resevoir during the upflow process. The reservoir is slightly over-pressured at the in- itiation of the instability (crack propagation) and it is this pressure diference (PI-P~) which does work against friction and may accelerate the fluid.

In the numerical simulation use has been made of for- mulation (2) which is thought to be physically more reasonable than case (1). Indeed, if PI=Pl everywhere along the ascent path, there would be little mechanical energy available to the fluid in overcoming friction, generating fractures and increasing the kinetic energy of the fluid.

The ratio of inertial to viscous forces in Eq. (3) is given by

DSp 2p Cy 6Z (7)

A short calculation reinforces the idea that for narrow, long cracks, viscous forces will dominate and fluid accelerations will generally be small. Precise ascent rates depend on crack roughness and tortuosity about which little is known. Pre- sumably these two parameters will change as metasomatic viens are produced.

The conservation-of-energy Equation (4) explicitly indicates how the temperature of the fluid varies in the vertical direction as the fluid expands and moves upward. It is con- venient to decompose the temperature equation and consider the physical aspects of the various terms. The third term on the RHS of (4) indicates that fluid tends to cool because of heat transfer between M M F within the crack and the surrounding lithisphere. The amount of cooling depends directly on the fluid-lithosphere temperature difference (T-Tw) and the heat-transfer coefficient, h and inversely on the width of the crack, initial mass transfer rate and isobaric heat capacity of the fluid. The importance of heat-transfer between fluid and surroundings may be noted by defining 0, the ratio of the isentropic (reversible, adiabatic) temperature gradient to the temperature gradient resulting from heat loss by heat-transfer. From (4) one finds

o - ~ g Y4 D T~ (8) 4h(Tf - Tw)

Note that an increase in either the fluid ascent rate or crack diameter decreases the importance of heat transfer, whereas a large value of h, (Appendix III) or A T favors heat transfer between M M F and surrounding lithosphere. Within the range of typical thermodynamic and transport parameters1, Eq. (8) indicates that heat transfer between M M F and the surrounding lithosphere can be significant and that the assumption of

l e=10-4K 2, D=lcm, T=I,500K, 21)/=102gm cm-2s -I, L =10km, kr 1 K - i s -1, AT=10K, Ce=2 x 107 erg gm -1K -1

i i r I t . 4 -

�9 i f , ~

1.2 I \ 9 0 0 ~ i .

! .0 ",,&

o8 %,,~-,... t 2 T o . E

o.o

0 . 2 900

I I I I 5 t 0 15 20

P R E S S U R E (Kb)

Fig. 1. Dashed and solid curves give the product c(T vs Pt for H20 and CO 2 at magmatic temperatures respectively, e is the isobaric expansivity and is defined according to c~_ -p-l(ctp/ctT)e. Sources of PVT are cited in the text

adiabatic flow is not necessarily justified a priori (i.e., often 0 < 1; cf. Kieffer and Delany 1979).

The other important thermal effect corresponds to decompression and ascent of fluid in the gravitational field. Decom- pression can lead to heating or cooling of the fluid depending on the thermodynamic path. Although reversible adiabatic isentropic expansion of an ideal gas will always lead to cooling, irreversible expansion of a real fluid may produce heating. This is related to the Joule-Thompson effect (Lewis and Randall 1961) and arises because of entropy production associated with irreversible expansion of a non-ideal fluid. Note that for an ideal gas, the first term in (4) vanishes (i.e., c~ = T -1 for ideal gas). For real fluids, the magnitude of the product ctT determines if heating occurs during irreversible adiabatic expansion. The term dPi/dZ is always negative; if c~T< 1 then the first term on the right hand side of (4) is positive and so heating may occur upon expansion. For adiabatic (h---0) flows in which inertial effects are unimportant (nearly incompressible flow), dT/dZ is positive (i.e., ascending fluid heats up) if

c ~ T - l d P f > g (9) py Cp dZ Cp

Smoothed values of c~T (see Spera and Bergman 1980 for the equations of state of U20 and CO2) have been plotted versus Pt at magmatic temperatures for H20 and CO2 (Fig. 1). Note that at high P both CO 2 and HzO have c~T<l; this means that irreversible decompression leads to a temperature rise. Also note that for H20, if P < 3 Kb then c~T> 1 and therefore irreversible adiabatic decompression is accompanied by cooling. For CO2, irreversible decompression invariably leads to heating. Assuming adiabatic (h=O) flow with c~T<I, Eq. (4) shows that an absolute maximum for the increase in fluid temperature due to viscous heating (entropy production) is

~T 2Cfl~/I2 ~Z pZ D Cp (10)

This effect arises because of the degradation of mechauical energy to heat when the fluid expands irreversibly. Adopting values appropriate for metasomatic fluid in a narrow crack (Cs=10 -a, /~ /=103gmcm - 2 s -z, ~SZ=10Km, D = l c m , py = l g m c m -3, C v = l . 0 c a l g m -1 K -1) one finds 5 T ~ 5 0 K . This shows that viscous heating may significantly increase the heat transport capabilities of migrating MMF. It is important

to emphasize, however, that this estimate of viscous heating is a maximum because the decrompression has been assumed to be entirely irreversible. The ratio of pressure decrements due to irreversible and reversible effects is given by

3 A P irreversible _2CfJ~I 2 (11) A P reversible pZ D g

This shows that high speed fluid moving through narrow, tourtnous and rough cracks can undergo significant irreversible decompressions. In most of our calculations however, c5 is rather small (6~10-~), and the increase of fluid temperature because of adiabatic irreversible decompression will be small (~ 10 to 50 K).

The results of many numerical simulations in which 6,/V/, Cr D, h and fluid compositions (H20 or CO2) were varied systematically show that the combined effects of conduc- tive/convective heat-transfer and temperature changes accom- panying decompression are often offsetting. This implies that ascending MMF rises approximately isothermally. This con- clusion is important because it shows that a corollary of MMF migration is the transport of mantle heat. In certain cases this heating may produce localized melting within the lithosphere. Bailey (1970) has suggested this possibility but on slightly different grounds than discussed here. Lang (1972) has also suggested the possibility of nearly isothermal magmatic flows.

As an example, consider a numerical experiment in which MMF migrates from a depth of 100kin to a depth of 50kin where, because of reaction and loss of pressure head, u f=0 . Detailed calculation shows that metasomatic fluid at temperature TI~I ,150~ invades a relatively cold (Tw~I,000~ piece of lithosphere. When thermal equilibrium between fluid and surroundings is attained, a small portion of lithosphere surrounding the crack will heat up. The temperature rise in the lithosphere is proportional to the diameter and number density of cracks, the duration of the metasomatic event and the temperature difference between mantle and fluid. The calculations presented in greater detail in Appendix II show that significant heating, perhaps leading to anatexis, is a natural consequence of the movement of MMF within the Earth.

Volatile Transport in the Earth: Thermal Implications

A question that arises from these considerations is the following: How important is the upward movement of heat by migrating MMF in terms of the thermal history of the Earth in general and the transient, localized heating of restricted portions of the lithosphere in particular? No unequivocal answer to the global question can be given because of un- certainty regarding the abundance, distribution and movement history of volatiles within the Earth. Additional un- certainty arises due to poorly constrained temperatures at depth and lack of information relative to the oxidation state of the upper mantle. The tectonic history of the Earth is also involved in that subduction of oceanic crust is a means whe- reby volatile recycling can occur.

Although we cannot offer a unique solution to these ma- jor geochemical problems, the role of MMF in heat-transfer on local scales can be discussed. Based on estimates of the amount of excess volatiles in the atmosphere, hydrosphere and in sedimentary rock reservoirs (e.g., see Walker 1977;

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Anderson 1975; Holland 1978; Pollack and Yung 1980; Li 1972) a~ least 2 x 1024gm of volatiles (mostly H20 and CO2) have been outgassed. Averaged over all of geologic time (4.55 x 109 year), this represents a minimum rate of volatile out-

gassing of 4.4 x 1014gm year -1. Given this rate and making a conservative estimate of the amount of thermal energy that MMF could transport based on the fluid dynamic model discussed earlier (see Appendix III), one finds that about 2 x 1016 cal year -1 can be transported by migrating fluids. This is sufficient to produce approximately 2x 1014gm year -~ of basaltic magma if the temperature of pre-metasomatised lithosphere is just at the solidus at the site of metasomatism. Although the implied magma production rate is about two orders of magnitude smaller than the present global rate, the volatile flux used in this computation is a minimum. Basic volcanic rocks of alkaline affinity account for a very small fraction of the Earth's annual magma production rate of 20Km 3 year -1. On this basis it seems reasonable to suggest that, at least for alkaline magmatism, heat transport by migrating MMF plays an important genetic role.

If copious degassing occured early in the history of the Earth (Fanale 1971; Walker 1976, 1977; Ozima 1973, 1975) the transport of deep mantle heat by MMF migration would have been an important heat-transfer process especially if volatiles were 'focused' into restricted regions. Indeed, New- ton et al. (1980) have argued that the stabilization of Archean crust against melting at its roots, and against resorption into a rapidly convecting mantle virtually requires that volatiles rich in the radiogenic elements were flushed upwards within 50 to 100 My of planetary accretion. Had this not been the case, in situ radiogenic heat production would have almost certainly led to partial fusion. Additionally, they note that a flux of CO 2 into the lower crust would lead to dehydration there. Because solidus temperatures in the crust increase markedly as aco2/an2o increases (for fixed P~ and bulk composition), a " C O 2 flush" would help stabilize the crust with tespect to anatexis. On a similar note, Schuiling and Kruelen (1979) have suggested that crustal metamorphic thermal domes may be heated by CO2-rich fluids from the mantle.

Conclusions

The important conclusions of this study may be summarized as follows:

1. A great deal of petrologic, isotopic and geochemical evidence supports the contention that low viscosity fluids rich in O - - H - - C dissolved silicates and the incompatible elements may ascend, decompress and act as metasomatic agents transporting heat and mass. These fluids may be important in generating isotopic heterogeneity in the mantle on the cm to km scale;

2. A mechanical constraint for upward migration of fluid is that fluid pressure (Pj.) be greater than or equal to the mean normal stress (or0) at any depth. Small differences in Pc-or o are sufficient to move low viscosity fluid to low pressure regions at geologically instantaneous rates;

3. The ordinary differential equations governing the vertical variation of pressure (P), temperature (T), velocity (u) and density (p) of a compressible, viscous, single-phase (H20 or CO2), non-reacting fluid ascending in a constant width crack have been solved. These calculations are thought to simulate the migration of mantle metasomatic fluid (MMF). The transport equations take into account irrevcrsibilities associated

62

with friction due to fluid flow through rough and tor tuous cracks and magma temperature changes due to heat-transfer and magma decompression;

4. The results of a large n u m b e r of numer ica l s imulat ions in which the various parameters governing flow behav ior were systematically varied (composi t ion of fluid, Cy, D, k, Tw(Z), rl, e, g), Re, Pr, Pc, initial condit ions) show the following:

(i) M M F tends to move at uniform velocities, on the order of m/s to 10m/s.

(ii) Tempera tu re of expanding M M F increases due to mechanical ly irreversible processes ( thrott l ing, skin-friction) and decreases due to reversible expansion and heat- t ransfer from M M F to sur rounding li thosphere. To a first approxi- ma t ion these processes ba lance one ano the r so tha t fluid ascends nearly isothermally (perhaps cooling a bit) from one level to another . A corollary of this behav ior is the t r anspor t of mant le heat to shallow and spatially restricted por t ions of the mantle. Numer ica l experiments (Appendix II) indicate tha t t r anspor t of heat into regions undergoing metasomat i sm can lead to anatexis.

5. A m i n i m u m est imate of the rate of the rmal energy t ranspor ted by mobi l ized volatile componen t s averaged over the age of the Ear th shows that the heat t ranspor ted can generate about 0.1 km 3 of basaltic magma per year. Al though this magma product ion rate is much smaller than the present day globally averaged rate of magma product ion (mainly MORB), it is of the correct order of magni tude for the product ion rate of alkaline magma;

6. Hea t t r anspor t by fluid ascent with concommi t an t metasomat i sm might have been an impor t an t mechan i sm early in the his tory of the Ear th if an early phase of copious degassing took place as suggested by Fana le (1971), Walker (1976), and Ozima (1975).

Acknowledgements. The work was supported by National Science Foundation grant EAR 78-03340. The computer skills of R. Gold and D. Roberson and discussions with Maggie Bergman and L. Noodles are gratefully acknowledged. Critical reviews by H.R. Shaw, S. Bergman, and A. Boettcher improved the form and content of this work.

Appendix I

Experimental data for the viscosity of geologically important volatile species at high Pt and T are sparce. However, existing experimental data for many simple fluids can be correlated fairly well by corresponding states methods (Hougen et al. 2946). Representative viscosities have been calculated for the species CH4, C02, HC1, H2S , HF, H20 at t=1,200~ and Pt=20kb and are presented below. Indi- vidual values vary by a factor of ten with a mean value of 7 x 10 .3 poise. HzS and CH 4 are the highest and lowest viscosity fluids respectively. These figures are crude estimates; however, they are

Component t/(1,200 ~ C, 20 kb) t/(2,200 ~ C, 20 kb) (poise) t/(1,200 ~ C, l bar)

HzS 1.3 x 10 .2 28 HC1 1 x 10 .2 20 HF 7 x 10 -3 14 C O 2 7 X 10 .3 13 H20 6 x 10 .3 13 CH 4 2 x 10 -3 4

likely to be accurate to at least an order of magnitude. If H20 and CO 2 are the dominant components of MMF, then a reasonable viscosity estimate for MMF at high Pt is 5 x 10 -3 poise. For compari- son, H20 , at STP has a viscosity of 10x 10 -3 poise. The ratio of fluid viscosity at 20kb and 1,200~ versus 1 bar and 1,200~ have also been tabulated. These data indicate that fluid viscosities will decrease by about an order of magnitude upon isothermal decompression. At low pressure, the viscosity of a gas increases with increasing T. It is expected that the viscosity of MMF will decrease with increasing T at high pressure.

Appendix II

The increase in lithospheric temperature at a fixed depth because of heat transport by MMF mobilization may be found by balancing the heat transferred by MMF (Q f) to that absorbed by the mantle undergoing metasomatism. Ignoring heats of reaction, conservation of energy requires that

Qm + QI = 0 (2a)

o r

~dp V(T- T)+Pc C{ Y/iAc z V(T- Tr (2b)

where T, T and Tr represent the final temperature of metasomatized mantle, the pre-metasomatized mantle temperature and the temperature of MMF respectively. All temperatures are evaluated at the depth Z where MMF upward migration ends. Other parameters are defined in the nomenclature. Rearrangement of (2b) gives

T - T a(Tr T) (2c) a+b

where a=IQrAcp c C~ and b=fiC e for the temperature rise of a piece of lithosphere undergoing metasomatism. As an illustrative calculation assume the following: pc= l per 50m 3, Ac=2cm 2, )1)/=3 x l 0 2 g m c m - 2 s -a, z=l day, CpY=2calgm-lK 1, ~p =0.3ca lgm -1 K -1, tS=3.3gm cm -3. Detailed numerical computation using. Eqs. (2) to (5) for pure CO2, assuming reasonable values for Cs, h, M~, and the geothermal gradient T.~(Z) and initiating fluid expansion from a depth of 100kin, shows that fluid migration ends at about Z = 75 km where T= 1,000 ~ C and T s = 2,150 ~ C. Equation (2c) indicates that the temperature increase in the volume (~10km 3) of mantle that is metasomatized is about 70~ The ratio of mass introduced by the metasomatic fluid to the pre-metasomatism mass is given by

)~l Ac z p~ (2d)

For the case considered here, the mass gain represents about 5 % of the total mass of mantle undergoing metasomatism. If the mantle was just at its solidus temperature at the time of metasomatism, partial fusion to the extent of 15 to 20wt.% will occur. This ignores the effect of MMF in lowering the peridotite solidus temperature and therefore represents a minimum.

Appendix III

Values of p, e, and fi were calculated using standard thermodynamic expressions and available PVT data for CO 2 and H20. In addition to references cited in the text, high P~-T free-energy data in Spera (1974) was utilized in predicting the properties of H20 at high P~-T. The high Pt data for H20 used here is consistent with that given in Delany et al. (1978) although their data were limited to low (sub- solidus) temperatures (t __< 800~ C). C~ for CO z and H20 were taken from Helgeson et al. (1978). The thermal conductivity and viscosity of H20 and CO 2 fluids were calculated as a function of Pt - T using the transport property tables in Hougen and Watson (1946). The skin-friction coefficient was calculated from the formulae

given below depending on the Reynolds number and pipe surface roughness:

16 c:=~e e

laminar flow, smooth crack Re<2.1 x 103,

C I =0.0791 Re -1/4

turbulent flow, smooth crack Re>2.1 x 103,

( I I~/2 ~ f ! = - 0.40 + 1.74 Ill (Re l /~f )

turbulent flow, rough crack Re>2.1 x 103.

The reader is referred to Goldstein (1938) and Bird et al. (1960) for a discussion of the relevance of these expressions. It should be noted that C s is a measure of skin-friction and surface roughness only; the effects of constrictions and bends (i.e., crack tortuosity) in increasing flow resistance have been approximately accounted for using expressions given by Bird et al. (1960), Chapter 7. The heat-transfer coefficient, h, depends on the length and width of the crack, Re, Pr and the thermal conductivity of the fluid. The correlation used to estimate h is taken from Kay (1968) and is

h= D Pe (3d)

where Pe ~-Re Pr and L is the effective length of the crack. Finally, T~,(Z), which represents the temperature along the crack wall as a function of depth (i.e., the lithospheric temperature distribution) is determined by assuming an arbitrary (but reasonable) temperature- depth distribution. The form

T,~(Z) =I% +/% Z 213 (3e)

was found useful, kt and k 2 are found by assuming a specific temperature at a given depth (e.g., t= 1,200 ~ C, Z = 100km).

Nomenclature

u, velocity, cm s -1 P, pressure, bar T, temperature, K t, temperature, ~ C:, fliction coefficient Ce, isobaric heat capacity, cal gm -1 K h, heat transfer coefficient, cal cm- 2 s-1 K 1 Z, vertical coordinate g, acceleration due to gravity, D, conduit diameter, cm L, length of conduit or crack X, mole fraction f / , mass flux, gm cm 2 s 1 Ac, cross-sectional area of single crack V, premetasomatism volume of mantle a~, activity of component i f~, fugacity of component i G, number density of metasomatic cracks P, density of mantle T~(Z), crack or volcanic conduit wall temperature as functon of depth k, thermal conductivity, cal cm- 1 s-~ K 1 Cp, isobaric heat capacity of mantle

Re, Reynolds number, puD _ M D

Pr, Prandtl number, Cp~/ k

uD Pe, Peclet n u m b e r , -

% duration of metasomatic event

~, isobaric expansivity, K - t fi, isothermal compressibility, bar -1 p, density, gm cm -3 t/, viscosity, (gm cm 1 s-1 =poise) e, pipe roughness, cm ~, ratio of irreversible to reversible pressure drop ~c, thermal diffusivity, cm 2 s -1

Subscripts

i, initial f, fluid l, lithostatic t, total m, magma

Superscripts

m, melt f, fluid

63

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Received January 26, 1981; Accepted May 4, 1981

carbon dioxide in igneous petrogenesis: ii. fluid dynamics of mantle metasomatism

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