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A Two-Porosity Double Lithology Model forPartial Melting, Melt Transport and Melt^rockReaction in the Mantle: Mass ConservationEquations and Trace Element Transport

YAN LIANG AND E. MARC PARMENTIER

DEPARTMENT OF GEOLOGICAL SCIENCES, BROWN UNIVERSITY, PROVIDENCE, RI 02912, USA

RECEIVED FEBRUARY 9, 2009; ACCEPTED NOVEMBER 19, 2009ADVANCE ACCESS PUBLICATION JANUARY 7, 2010

To better un derstand the dyna mical processes of partial melting, melt

transport, and melt^rock reaction in the mantle, mass conservation

equations for a two-porosity double lithology continuum have been

developed using the method of volume averaging. Here the region of

interest is treated as two overlapping continua occupied by a low-

porosity peridotite matrix and a high-porosity dunite channel net-

work. Conservation equations for the matrix and channel continua

are coupled through mass transfer terms that include matrix dissolu-

tion and diffusive and advective mixing between the melt in the chan-

nel and that in the matrix. Essential features of the two-porosity

double lithology model have been investigated through simple case

studies. In general, the composition of the channel melt is a weighted average of the matrix melt extracted along the melting column (as a

result of melt suction) and the matrix materials dissolved into the

channel melt (as a result of matrix^channel transformation).

Dissolution of pyroxene in the peridotite matrix into the channel

melt and precipitation of olivine lowers the compatible trace element

abundances in the channel melt. In the absence of matrix dissolution,

the channel melt is equivalent to the aggregated or pooled matrix

melt. The incompatible trace element abundance in the channel melt

is dominated by the less depleted small-degree melts from the lower

part of the melting column and not very sensitive to the details of

how the melt suction rate and matrix^channel transformation rate

vary spatially. The incompatible trace element abundances in the

matrix melt and solid are very sensitive to the direction of melt flowacross the matrix^channel interface, the magnitude and variation of

the relative melt suction rate, and the depth of dunite channel initia-

tion, and depend moderately on variations in channel volume fraction

in the double lithology region. Percolation of the enriched channel

melt into the more depleted residual matrix in the upper part of the

melting column may provide a viable mechanism for late-stage melt

refertilization or mantle metasomatism. Understanding the

first-order characteristics of the channel and matrix melts and

solids is essential in deciphering the melting and melt migration his-

tories of residual mantle rocks and erupted basalts.

KEY WORDS: double porosity; double lithology; mantle melting; dunite

channel; harzburgite; trace element; metasomatism; melt^rock reaction;

melting model; reactive dissolution

I N T R O D U C T I O NThe generation and segregation of melt from the Earth’sinterior is a fundamental, but only partially understood,process. Segregation of melt involves two-phase flow inwhich low-viscosity melt percolates through a much moreviscous solid matrix (e.g. Sleep, 1974; McKenzie, 1984;Scott & Stevenson, 1986; Bercovici et al ., 2001). As illu-strated in Fig.1, melting and melt segregation involve inter-acting physical processes on a wide range of length scales.Melt is generated at the millimeter scale of single mineralgrains (grain scale). This melt, driven by buoyancy forces

and stresses in the solid, migrates through a porous orchannelized continuum that forms at scales of tens of meters (transport scale). The large-scale setting in whichmelt is generated involves solid mantle flow and melt-ing on scales of hundreds of kilometers (tectonic scale).

Corresponding author. E-mail: [email protected]

The Author 2010. Published by Oxford University Press. Allrights reserved. For Permissions, please e-mail: [email protected]

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This is illustrated in the figure by a diverging plate bound-ary at which tectonic plates move apart with hot, largelysolid mantle upwelling and melting to generate the basalticoceanic crust that makes up nearly all of the Earth’sseafloor.

One fundamental geological observation is that the

chemical composition of erupted basalt is not in equilib-rium with residual mantle at low pressure (e.g. O’Hara,1965; Stolper, 1980; Elthon & Scarfe, 1984; Klein &Langmuir, 1987; Kinzler & Grove, 1992), particularly atdiverging plate boundaries. To preserve the geochemicalsignature developed at depth, melts must rise from depthsof at least 30 km to the surface without extensivere-equilibration with their surrounding mantle at shallowdepth. This can be accomplished by focused melt flowthrough melt-filled fractures or high-permeability porouschannels or a combination of the two (e.g. Sleep, 1988;Nicolas, 1989, 1990; Aharonov et al ., 1995, 1997; Kelemenet al ., 1995a, 1995b, 1997; Spiegelman et al ., 2001). Several

lines of evidence suggest that the porosity and lithologystructures in the melt generation and extraction region of the mantle are heterogeneous, consisting of interconnectedhigh-porosity dunite channels embedded in a low-porosityharzburgite or lherzolite matrix (Kelemen et al ., 1997, andreferences therein; also see Fig. 1). The formation of thedunite channels in the mantle involves reactive dissolutionas olivine normative basalts percolate through a harzbur-gite or lherzolite matrix (e.g. Quick, 1981; Kelemen, 1990;Kelemen et al ., 1992, 1995a, 1997; Daines & Kohlstedt,1994; Asimow & Stolper, 1999; Morgan & Liang, 2003,2005; Beck et al ., 2006). Preferential dissolution of pyroxeneand precipitation of olivine increases local porosity and

grain size (Daines & Kohlstedt, 1994; Morgan & Liang,2003, 2005; Braun, 2004), and hence dunite permeability,which accelerates melt flow and peridotite dissolution.Such positive feedbacks between dissolution and melt flowresult in reactive infiltration instabilities and the develop-ment of elongated dunite channels along the melt flowdirection (e.g. Ortoleva et al ., 1987; Steefel & Lasaga,1990;Daines & Kohlstedt, 1994; Aharonov et al ., 1995, 1997;Kelemen et al ., 1995b; Spiegelman et al ., 2001).

Remnants of dunite channels have been observed asveins, tabular or sometimes irregular-shaped bodies inophiolites where mantle rocks are exposed at the surface.According to Kelemen et al . (1997), these dunite channelsmake up 5^15% of the mantle in the Oman ophiolite andtheir sizes range from tens of millimeters to 200m inwidth and tens of meters to at least 10 km in length. Themajority of the dunite channels have sharp lithologicalcontacts with their surrounding harzburgite or lherzolitematrix, forming a distinct dunite^harzburgite or dunite^harzburgite^lherzolite sequence in the field. A power-lawrelationship between the frequency of the dunite bodiesand their width was found through detailed field studies,

and this relationship has been used to infer the melt fluxthrough the dunite channels (Kelemen et al ., 2000; Braun& Kelemen, 2002). Although their spatial distribution inthe mantle is still not well constrained, an emerging pic-ture is that a high-porosity, interconnected, coalescing net-work of dunite channels is likely to exist above or within

the melting region in the mantle (Fig. 1). It has been sug-gested that localized high-permeability dunite channelsproduced by peridotite reactive dissolution allow melt tosegregate efficiently from its source region while preservingits geochemical signature developed at depth (e.g.Kelemen et al . 1997, and references therein). Channelizedporous flow is also capable of producing uranium-seriesdisequilibrium and significant trace element fractionationduring mantle melting and melt extraction (e.g. Iwamori,1993, 1994; Spiegelman & Elliott, 1993; Kelemen et al .,1997; Lundstrom, 2000; Jull et al ., 2002; Spiegelman &Kelemen, 2003).

One of the most interesting petrological and geochemi-

cal observations of dunite and harzburgite from themantle sections of ophiolites is the presence of concentra-tion gradients in major, minor, and trace elements acrossthe dunite^harzburgite or dunite^harzburgitê lherzolitesequence (e.g. Quick, 1981; Obata & Nagahara, 1987;Kelemen et al ., 1992; Takahashi, 1992; Takazawa et al .,2000; Kubo, 2002; Suhr et al , 2003; Braun, 2004;Lundstrom et al ., 2005; Zhou et al ., 2005; Morgan et al .,2008). Development of the concentration gradients involvesat least the following mass transfer processes between thechannel and the matrix: (1) transformation of melt-bearingharzburgite into melt-bearing dunite as a result of dissolu-tion of orthopyroxene and precipitation of olivine; (2) dif-

fusive and dispersive exchange between the melt in thedunite channel and the melt in the matrix; (3) advectivemixing between the melt in the channel and the melt inthe matrix (e.g. Morgan & Liang, 2003, 2005; Spiegelman& Kelemen, 2003; Lundstrom et al ., 2005; Morgan et al .,2008). Diffusive, dispersive, and advective exchangesbetween the melt in the channel and the melt in thematrix are supported by the smooth (rather than sharp)and frequently asymmetric concentration profiles in min-erals across the dunite^harzburgite interface in the field(e.g. Kelemen et al ., 1992; Suhr et al ., 2003; Braun, 2004;Morgan et al ., 2008). Although the amount of materialadded to the dunite channel has been assumed to be very

small, the extent of mass transfer between the peridotitematrix and the melt flowing through the dunite channel isstill not known. One of the main objectives of this study isto identify key parameters controlling the extent of masstransfer between the melt in the channel and the materialsin the matrix.

Although significant progress has been made in under-standing the nature and consequences of channelizedmelt flow in the mantle (Spiegelman & Kenyon, 1992;

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Hart, 1993; Iwamori, 1993, 1994; Spiegelman & Elliott,1993; Daines & Kohlstedt, 1994; Aharonov et al ., 1995,1997; Kelemen et al ., 1995b, 1997; Lundstrom, 2000;Ozawa, 2001; Jull et al ., 2002; Morgan & Liang, 2003,

2005; Spiegelman & Kelemen, 2003; Katz, 2008; Liang,2008), a quantitative model for partial melting, melt migra-tion, and melt^rock reaction in a two-porosity doublelithology mantle is still not fully developed. A starting

channel

continuum

Exchange

D

Outcrop scale ~10 m

α, S, Mcm.

melting

Γ m

k m, φ

m, V

f

m, V

s

m, C

f

m k ch, φ

ch, V

f

ch, V

s

ch, C

f

ch

matrix

continuum

H

Tectonic scale ~10-100 km(a)

(b)

(c)

0 X b (t ) x

z

single

double

L

h

z d

Fig. 1. (a) Schematic diagram illustrating mantle flow and melt migration on the tectonic scale in a mid-ocean ridge setting where the shadedtriangular area represents regions of the mantle that are partially molten. The melting triangle consists of a lower one-porosity single lithologyregion (marked as ‘single’) and an upper two-porosity double lithology region (labeled as ‘double’). The region for the 1-D models discussed in

the text is highlighted by the white dashed rectangle. (b) A close-up view of the melt migration region of the mantle that consists of alow-porosity harzburgite or lherzolite matrix and an interconnected higher porosity and larger grain size dunite channel network. (c) An idea-lized dunite^harzburgite cross-section where mass, momentum, and heat are transported both parallel and perpendicular to the dunite^harzburgite contact (arrows). In the local coordinate space, x ¼ 0 corresponds to the center of the harzburgite strip and x ¼ L corresponds tothe center of the dunite channel. X b(t ) marks the dunite^harzburgite boundary.

LIANG & PARMENTIER A TWO-POROSITY DOUBLE LITHOLOGY MODEL

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point in any quantitative study of melt migration in atwo-porosity, double lithology mantle on the tectonic scale(Fig. 1) is a set of mass, momentum, and energy conserva-tion equations that take into account the processes of soliddeformation, matrix melting, dunite channel formation,and chemical exchange between the melt in the channel

and the melt in the matrix. In principle, one can includechannel formation processes into the tectonic-scale meltmigration models by explicitly modeling every channelusing conservation equations originally developed for flowin a simple, deformable, porous medium (e.g. McKenzie,1984; Aharonov et al ., 1995; Spiegelman et al., 2001;Spiegelman & Kelemen, 2003). In practice, one quicklyruns into computational limitations such as resolving fine-scale evolving (fractal tree-like) structures that have beenenvisioned to develop and that are difficult to resolvenumerically.

A double porosity model, first developed for large-scaletransport problems in fractured rocks in hydrology (e.g.

Barenblatt et al ., 1960), provides one possibility for movingbeyond treating each channel individually. Here theregion of interest is treated as two ‘overlapping’ continuaoccupied by the low-porosity matrix and interconnectedhigh-porosity channel system (Fig. 1). The physical andchemical properties of the channel continuum and matrixcontinuum at a given point are defined in a local represen-tative elementary volume (REV) that is larger than a typ-ical channel width or spacing but smaller than thetectonic length-scale of interest such that channel-scalefluctuations are smoothed out over the tectonic scale (e.g.Bear, 1993; Quintard & Whitaker, 1996; see also Fig. 1).The double-porosity concept has already been discussed

or proposed for magma transport beneath mid-oceanridges (e.g. Iwamori, 1993, 1994; Spiegelman & Elliott,1993; Kelemen et al ., 1997; Lundstrom, 2000; Ozawa, 2001;

Jull et al ., 2002; Liang, 2008; also see Spiegelman &Kelemen, 2003). Although promising, these first generationmodels ignore channel formation, channel spatial distribu-tion, as well as channel^matrix diffusive and dispersiveexchange. One of the primary objectives of this study is todevelop a set of mass conservation equations that can beused to study the chemical exchanges (i.e. channel forma-tion, channel^matrix diffusive and advective exchange)between the melt and solid in the channel and those inthe matrix.

In this study we expand early double porosity models forfluid and melt migration to include channel formationand channel^matrix chemical exchange. For simplicity weconsider two overlapping porous media that consist of twodistinct lithologies: a high-porosity dunite continuum anda low-porosity harzburgite continuum (hence the phrase‘two-porosity double lithology’). In the next section, wepresent the mass conservation equations for a two-porositydouble lithology medium in which the interstitial melt

and solid matrix are in local thermodynamic equilibrium.(Momentum and heat transfer between the channel andmatrix continua can be formulated in a similar way andwill be discussed in a future publication.) Using one-dimensional (1-D) steady-state models, we explore howthe incompatible and compatible trace element abun-

dances in the melt in the channel and the matrix continuarespond to variations in channel volume fraction and meltsuction rate in the upwelling melting column. We willshow how the depth of dunite channel initiation affectsthe extent of depletion of the incompatible trace elementsin the channel melt and the matrix melt, how the direc-tions of melt flow across a matrix^channel boundaryaffects the extent of enrichment of the incompatible traceelements in the matrix, and why dissolution of pyroxeneand precipitation of olivine as dunite channels formlowers the compatible trace element abundance in thechannel melt. Although flow only into dunite channels issometimes assumed or implied, the discharge of enriched

channel melt into the more depleted residual matrix inthe upper part of the melting column may result in themelt refertilization or mantle metasomatism that has beencalled upon to explain light rare earth element (LREE)-enriched patterns in residual peridotites. We conclude thiscontribution with a discussion of potential geochemicalapplications. In Appendices A and B, we present a simpleprocedure for identifying the various source and sinkterms in the conservation equations and equations for the1-D steady-state models used in our analysis in the maintext.

M A S S C O N S E R V A T I O NE Q U A T I O N S

Three sets of mass conservation equations are needed foran overlapping two-porosity double lithology system: (1)conservation equations for the channel and matrix con-tinua; (2) conservation equations for the melt and solidwithin the channel and matrix continua; (3) conservationequations for a chemical species in the channel andmatrix continua. We derive these equations sequentiallybelow.

Consider a two-porosity double lithology medium con-sisting of high-porosity interconnected channels of onelithology (channel continuum) and a low-porosity matrixof a different lithology (matrix continuum). FollowingBarenblatt et al. (1960), we define the physical and chemicalproperties of the interstitial melt and the solid in the chan-nel continuum and the matrix continuum over a localREV through volume averaging (e.g. see also Bear &Bachmat, 1990; Bear, 1993; Quintard & Whitaker, 1996;Drew & Passman, 1999; Whitaker, 1999). The concentra-tions of a chemical species in the melt at a given point (x)


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in the channel and matrix continua, C ch f and C m

f , for exam-ple, are defined as

C ch f x,t ð Þ ¼ 1

Xch f

Z Xch f

Ĉ ch f ~ x,t ð ÞdX

and

C m f x,t ð Þ ¼ 1

Xm f

Z Xm f

Ĉ m f ~ x,t ð ÞdX

where X ch f and Xm

f are the volumes of melt in the channelcontinuum and the matrix continuum within the REV,respectively; Ĉ ch f ~ x,t ð Þ and Ĉ

m f ~ x,t ð Þ, functions of local posi-

tion (~ x) within the REV, are the transport-scale concentra-tions of the element of interest in the melt within thechannel and matrix continua, respectively. Other physicaland chemical properties are defined in a similar fashion.

Total mass conservation equations for the

channel and matrix continuaIt is convenient to define the mean density and velocity forthe channel and matrix continua in the REVas follows:

rch rch

f fch þ rchs 1 fch

,rm rm

f fm þ rms 1 fm

Vch rch f fchV

chf þ r

chs 1 fch

Vchs

rch,

Vm rm f fmV

mf þ r

ms 1 fm

Vms

rm

where r is the density; f is the porosity or volume fractionof the melt; V is the barycentric velocity; the subscripts or

superscripts ch, m, f , and s refer to channel, matrix, melt(or fluid), and solid, respectively. Vchf , for example, is themelt velocity in the channel continuum within REV.Symbols used in this study are listed in Table 1.

Changes in total mass of the channel continuum in theREVare due to flow of channel materials across the exter-nal boundary of the REV and sources or sinks within theREV. For melt migration and melt^rock interaction in themantle, two source or sink terms are important: (1) trans-formation of the matrix into the channel (e.g. melt-bearingharzburgite into melt-bearing dunite), measured by a rateM cm (mass per unit volume of REV per unit time); (2)flow of melt from the matrix to the channel continuum

within the REV, characterized by a rate _

S (mass per unitvolume of REV per unit time). The melt suction rate _S isnegative when melt flows from the channel into thematrix. It should be noted that the melt suction ratedefined above is slightly different from the melt suctionrate of Iwamori (1993), Lundstrom (2000), and Jull et al .(2002). The latter was defined as mass per unit volume of the matrix (rather than per unit volume of the REV) perunit time.

Matrix^channel transformation or channel formationhas not been included in previous double-porosity models.Taking w as the volume fraction of channel within REV,the overall mass conservation equations for the channeland matrix continua can be derived using Reynolds’ trans-port theorem (e.g. Aris, 1962); namely,

@ rchc @t

þ r rchcVch

¼ M cm þ _S ð1aÞ

@ rm 1 cð Þ

@t þ r rm 1 cð ÞVm

¼ M cm _S ð1bÞ

where t is time. The advantage of defining M cm and _S interms of mass per unit volume of the REV, rather massper unit volume of the matrix, is evident by the simpleexpressions of (1a) and (1b) [see also equations (2a)^(3b)below].

The volume fraction of the channel can be determinedby the matrix^channel transformation rate M cm. The

channel volume fraction w is related to the matrix^channeltransformation rate by the simple relationship

@c

@t þ Vm r c ¼

1 cð ÞM cmrm

ð1cÞ

where the left-hand side (LHS) of equation (1c) is thematerial derivative following the moving matrix(solid þ melt); the factor (1 ^w)/rm on the right-hand side(RHS) of equation (1c) accounts for the fraction of matrixavailable for dissolution at the given time and position.Equation (1c) is, in spirit, similar to the evolution equationgoverning the degree of melting experienced by the solidmatrix [see equation (8d) below].

Conservation equations for the melt andsolid within the channel and matrixcontinuaMass conservation equations for the melt and solid withinthe channel continuum and the matrix continuum, respec-tively, can be obtained once the source and sink terms arespecified. Creation of channels within the REV will redis-tribute melt and solid between the matrix and the channel,at the rates of wchM cm and (1 ^ wch)M cm for the melt andsolid, respectively, within the channel continuum, wherewch is the average weight fraction of the melt in the channel

continuum in the REV. If

ch and

m are the melting ordissolution^precipitation rates within the channel andmatrix continua [measured as mass per unit volume of the channel or matrix per unit time, as defined byMcKenzie (1984)], respectively, the overall rates atwhich melt and solid are created in the channelaveraged over the REV are cch þ wchM cm þ _S andcch þ 1 wchð ÞM cm, respectively, whereas the overallrates at which melt and solid are created in the matrix are


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1 cð Þm wmM cm _S and 1 cð Þm 1 wmð ÞM cm,

respectively, where wm is the average weight fraction of the melt in the matrix continuum in the REV. Hencemass conservation equations for the melt and solid withinthe channel continuum take the form

@ rch f fchc

@t þ r rch f fchcV

chf

¼ cch þ wchM cm þ _S

ð2aÞ

@ rchs 1 fch c @t

þ r rchs 1 fch cVchs ¼ cch þ 1 wchð ÞM cm:

ð2bÞ

Similarly, mass conservation equations for the melt andsolid within the matrix continuum can be written as

@ rm f fm 1 cð Þh i

@t þ r rm f fm 1 cð ÞV

mf

h i¼ 1 cð Þm wmM cm _S

ð3aÞ

Table 1: List of symbols used in the main text

Symbol Meaning Units

C ch

f , C m

f element concentration in the melt in the channel or matrix continuum

C ch

s , C m

s element concentration in the solid in the channel or matrix continuum

Ĉ ch

f , Ĉ m

f transport-scale concentration of an element in the channel or matrix

Dch, Dm dispersion coefficient for a component in channel melt or matrix melt m2

/s

Da melting Damko ¨ hler number, equation (10h) m2

/s

F degree of the melting experienced by the solid matrix, equation (8d)

h height of the melting column m

k ch, k m solid–melt partition coefficient for the channel or matrix continuum

k E

ch, k E

m effective partition coefficient for the channel or matrix continuum, equation (8c)

M cm matrix–channel transformation rate kg/m3

s

Qcm mass transfer rate as a result of matrix–channel transformation, equation (4a) kg/m3

s

Qd channel–matrix diffusive exchange rate, equation (5) kg/m3

s

Qs channel–matrix advective mixing rate, equation (6) kg/m3 s

R 0S

dimensionless melt suction rate, equation (10h)

_S melt suction rate kg/m3

s

Vch, Vm velocity of bulk channel or matrix continuum m/s

Vchf , Vmf velocity of melt in the channel or matrix continuum m/s

Vchs , Vms velocity of solid in the channel or matrix continuum m/s

w ch, w m weight fraction of melt in the channel or matrix continuum in REV

x tectonic-scale spatial coordinate m

~ x channel-scale spatial coordinate m

z tectonic-scale vertical coordinate m

z d height of the single lithology column m

a coefficient for channel–matrix diffusive exchange, equation (5) kg/m3 s

b, , a coefficients for matrix–channel tra nsformation rate and melt suction rate, equations (9a) and (9b) kg/m3

s

fch, fm porosity of channel or matrix continuum

ch melting or dissolution–precipitation rate of channel continuum kg/m3

s

m melting rate of matrix continuum kg/m3 s

0m

re-scaled melting rate of matrix continuum, 0m

¼ ð1 cÞm kg/m3 s

dimensionless diffusive and dispersive exchange rate, equation (10h)

S dimensionless melt suction rate, equation (18d)

H dimensionless matrix–channel transformation rate, equation (18d)

Xchf

, Xms

volume of melt in the channel or matrix continuum in REV m3

rch, rm mean density of channel or matrix continuum kg/m3

rchf

, rchs

density of melt or solid in the channel continuum kg/m3

rmf , rm

s density of melt or solid in the matrix continuum kg/m3

w volume fraction of channel in REV, equation (1c)

w0 characteristic channel volume fraction, equation (10h)


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@ rms 1 fm

1 cð Þ

@t þ r rms 1 fm

1 cð ÞVms

¼ 1 cð Þm 1 wmð ÞM cm:

ð3bÞ

In Appendix A, we outline a simple procedure that can beused to identify the various source and sink terms in equa-tions (2a)^(3b). The overall mass conservation equationfor the channel continuum [equation (1a)] or the matrixcontinuum [equation (1b)] is recovered if we sum equa-tions (2a) and (2b) or equations (3a) and (3b).

Conservation equations for achemical speciesTo obtain the mass conservation equation for a chemi-cal species in the channel and matrix continua, weassume that the solid and melt are in local chemicalequilibrium within the channel and matrix continuain the REV, respectively. Hence for a trace element

of interest, its concentration in the bulk solid (C s) inthe channel or matrix can be calculated from the respec-tive melt composition (C f ) through the equilibriumpartition coefficient k; namely, C chs ¼ kchC

ch f and

C ms ¼ kmC m

f . The total mass densities of a trace elementin the channel and matrix continua in the REVare then given by ½rch f fch þ r

chs 1 fch

kchcC ch

f and½rm f fm þ r

ms 1 fm

km 1 cð ÞC m f respectively, where C ch

f

and C m f are the concentrations (in weight fraction) of thecomponent of interests in the melt in the channel andmatrix continuum, respectively, in the REV. In the absenceof radioactive decay, changes in the mass of a component inthe channel or matrix in the REVare due to (1) advection

of both the solid and the melt across the external boundaryof the REV, (2) hydrodynamic dispersion in the melt acrossthe boundary of the REV, and (3) chemical exchangesbetween the channel and matrix continua within the REV.In general, hydrodynamic dispersion in a porous mediumis due to both chemical diffusion in the melt and grain-scalemelt flow in the porous medium (e.g. Bear & Bachmat,1990; Phillips, 1991; Whitaker, 1999). The dispersive andadvective fluxes across the REV boundary have been trea-ted previously (e.g. McKenzie, 1984; Bear & Bachmat,1990; Phillips, 1991; Whitaker,1999). Here we will focus onthe matrix^channel chemical exchange rates.

Based on geological field observations and laboratory

reactive dissolution studies, at least three sources contrib-ute to the chemical exchange between the channel and thematrix: (1) transformation of the matrix into the channelby matrix dissolution at the channel^matrix interface; (2)diffusive and dispersive mixing between the melt in thechannel and the melt in the matrix; (3) advective mixingbetween the melt in the channel and the melt in thematrix. Diffusive exchange between the solid in thematrix and the solid in the channel is negligibly small

compared with the three terms listed above and will notbe considered in this study. Because the bulk (solid þ melt)matrix converts to the bulk channel continuum at a rateM cm (>0), the rate at which a chemical species transfersfrom the matrix to the channel is

Q cm ¼ wmC m

f þ 1 wmð ÞC m

sh i

M cm: ð4aÞ

For a trace element equation (4a) can be simplified in thelimit of local chemical equilibrium between the melt andcrystals in the matrix within the REV,

Q cm ¼ wm þ 1 wmð ÞkmC m

f M cm:h

ð4bÞ

The rate of diffusive and dispersive mixing in the meltsbetween the channel and the matrix across the channel^matrix interface within the REV must be a function of the differences in composition between the melt in thechannel and the melt in the matrix. As an approximation

we assume a linear exchange rate

Q d ¼ A C m

f C ch

f

ð5Þ

where a is the channel^matrix diffusive exchange coeffi-cient, measured in mass per unit volume of REV per unittime; A is the fraction of channel^matrix interfacial areaavailable for diffusive transport. In this study we assumeA ¼w.

Advective mixing between the melt in the matrix andthat in the channel depends on the direction of melt flowacross the matrix^channel boundary. Let _S be the rate atwhich melt flows from the matrix into the channel and Q s

be the average mass flux of a component of interest as aresult of melt flow across the matrix^channel interface inthe REV. We have Q s ¼ C m f _S when melt flows from thematrix to the channel (_S >0), and Q s ¼ C ch f _S when meltflows from the channel to the matrix (_S


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@ rm f fm þ rms 1 fm

km

h i 1 cð ÞC m f

n o@t

þr rm f fmVmf þ r

ms 1 fm

kmVms

h i 1 cð ÞC m f

n o¼ r rm f fm 1 cð ÞDmr C

m f h i Q d Q s Q cm

ð7bÞ

where Dch and Dm are the tectonic-scale hydrodynamicdispersion tensors for the element of interest in the channelcontinuum and matrix continuum, respectively. Thesecond terms on the LHS of equations (7a) and (7b)account for melt and solid advection in the channel andmatrix continuum, respectively, whereas the first terms onthe RHS of equations (7a) and (7b) account for dispersionin melt within the channel and matrix continuum, respec-tively. The tectonic-scale hydrodynamic dispersion tensoris a sum of chemical diffusivity in the melt and mechanicaldispersivity as a result of grain-scale melt flow in the

porous media, and hence is direction dependent (fordetails see Bear & Bachmat, 1990; Phillips, 1991; Whitaker,1999, among others). The second and third terms on theRHS of equations (7a) and (7b) (Q d and Q s) arise fromtransport-scale dispersive and advective mixing betweenthe melt in the channel and that in the matrix, and thelast terms are due to dissolution of the matrix (M cm> 0).With the help of equations (2)^(6), equations (7a) and(7b) can be expanded in non-conservative but familiarforms,

rch f fch þ rchs 1 fch

kch

h ic@C ch f

@t

þ rch f fchVchf þ rchs 1 fch

kchVchsh i

c r C ch f

¼ r rch f fchcDchr C ch

f

h iþ c kch 1ð ÞC

ch f ch

þ c þ _S þ _S =2 C m f C ch f þ kE mC m f kE chC ch f M cm

ð8aÞ

rm f fm þ rms 1 fm

km

h i 1 cð Þ

@C m f

@t

þ rm f fmVmf þ r

ms 1 fm

kmVms

h i 1 cð Þ r C m f

¼ r rm f fm 1 cð ÞDmr C m

f

h iþ 1 cð Þðkm 1ÞC

m f m

ac _S _S

=2

C m f C ch f

ð8bÞ

where _S is the absolute value of the melt suction rate; kE m

and kE ch are the bulk or effective matrix/melt and channel/melt partition coefficients for the element of interests,namely,

kE m ¼ wm þ 1 wmð Þkm and kE ch ¼ wch þ 1 wchð Þkch: ð8cÞ

Finally, the degree of melting experienced by the solidmatrix (F ) is an important variable in mantle melting stu-dies. It is defined with respect to the moving solid matrix,namely,

@F

@t þ Vms r F ¼

1 F ð Þmrms 1 fm ð8dÞ

where the factor 1 F ð Þ=rms 1 fm

on the RHS of equa-tion (8d) accounts for the fraction of solid matrix availablefor melting (Ichihara & Ida,1998; Liang, 2008). Equations(1)^(3) and (8) form a complete set of mass conservationequations for modeling partial melting, melt transport,and melt^rock reaction in a deforming two-porositydouble lithology mantle when radioactive decay is absentand local chemical equilibrium between the melt and thesolid in the channel and the matrix continuum is estab-lished, respectively, within the REV. They are the startingpoint of our subsequent studies of the physical and chemi-cal consequences of melt generation and segregation in a

multi-scale heterogeneous mantle.

E S S E N T I A L F E A T U R E S O F T H E

T W O - P O RO S I T Y D O U B L E

L I T H O L O GY M O D E LOne of the key features of the two-porosity double lithol-ogy model outlined in the preceding section is the couplingor chemical and hydrodynamic interactions between thechannel continuum and the matrix continuum. In theabsence of channel^matrix interaction, equations (2), (3),and (8) reduce to the standard mass conservation equa-

tions for partial melting and melt migration in aone-porosity single lithology porous medium in which thecrystalline matrix is in local chemical equilibrium with itsinterstitial melt (e.g. McKenzie, 1984). Although the chem-ical consequences of partial melting, melt transport, andmelt^solid chemical exchange in a one-porosity singlelithology mantle are relatively well understood (e.g.McKenzie, 1984, 1985, Ribe, 1985a, 1985b; Richter, 1986;Navon & Stolper, 1987; Iwamori, 1993; Qin, 1993;Richardson & McKenzie, 1994; Spiegelman, 1996), theessential features of partial melting, melt transport, andmelt^rock reaction in a two-porosity double lithologymantle are not, in spite of recent progress (e.g. Iwamori,

1993, 1994; Spiegelman & Elliott,1993; Asimow & Stolper,1999; Lundstrom, 2000; Ozawa, 2001; Jull et al ., 2002;Liang, 2008; also see Spiegelman & Kelemen, 2003). Forexample, the roles of matrix^channel transformation andchannel distribution on the compatible and incompatibleelement abundances in the matrix and channel melts arestill not known. The relationship between trace elementabundance in the channel melt and that in the matrix isstill not well understood.


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In this section, we consider a simple problem of concur-rent melting, melt migration, and melt^rock reaction in a1-D steady-state upwelling two-porosity double lithologymantle column. This problem is directly relevant to melt-ing beneath a mid-ocean ridge and around the central por-tion of a mantle plume where the streamlines of the melt

and the solid are parallel or sub-parallel (region enclosedby the white dashed rectangle in Fig. 1a). From phase equi-librium and petrological considerations, we divide thesteady-state melting column into two regions: a singlelithology lower region and a double lithology upperregion (Fig. 1a). In the absence of horizontal melt flow,Ribe (1985b) showed that the trace element abundance inthe melt in the single lithology region obeys the batch melt-ing model. Iwamori (1993), Ozawa (2001), Liang (2008),and Liang & Peng (2009) gave analytical expressions forthe abundance of a trace element in the melt and solid inthe double lithology region when the channel volume frac-tion is constant and uniform and the rate of matrix^chan-

nel transformation is zero. To highlight the new features of the two-porosity double lithology model developed in thisstudy, we focus on the variations of the compatible andincompatible trace elements in the double lithology upperregion. For simplicity, we neglect dissolution^reprecipita-tion within the channel (ch ¼ 0) and diffusive and disper-sive transport along the vertical direction in the tectonicscale.

One-dimensional steady-state equationsWe are interested in the variations of the channel volumefraction (w), degree of melting experienced by the solidmatrix (F ), and the melt compositions in the channel andthe matrix (C ch f and C

m f ) in a 1-D upwelling steady-state

two-porosity double lithology column (Fig. 1a). For pur-

pose of illustration, we consider a spatially variablematrix^channel transformation rate and melt suction rategiven by

M cm ¼ M 0M zð Þ ¼ M 0z zd

h

bfor z > zd ð9aÞ

_S ¼ S 0S zð Þ ¼ S 0 1 þ a z zd

h

for z > zd ð9bÞ

where M 0, S 0, a, b, and are prescribed constants; M (z)and S (z) are the dimensionless dissolution rate and themelt suction rate, respectively; zd is the height of the singlelithology column; and h is the height of the melting column.

To facilitate comparison with previous studies, we assumethat the scaled matrix melting rate 1 cð Þm ¼ 0m is con-stant and uniform in the melting column. Figure 2 displaysexamplesof the normalizedmatrix^channeltransformationrates and melt suction rates. Also, for convenience, we scalethe vertical coordinate (z) by the height of the meltingcolumn (h). The nondimensionalized equations and theboundary conditions are

dC m f dz

¼Da km 1ð ÞC

m f C

m f C

ch f

c 12 R

0S R

0S

S zð Þ F d þ 1 F d ð Þkm þ Da 1 kmð Þ z zd ð Þ R

0S I S c0k

E mI M Þ

ð10aÞ

dC ch

f

dz¼ C

m

f C

ch

f

c þ

1

2 R

0

S þ R

0

S

S zð Þ

þc0 k

E

mC

m

f k

E

chC

ch

f

M zð ÞR0S I S þ c0k

E chI M

ð10bÞ

dF

dz¼

Da 1 F ð Þ

1 F d Da z zd ð Þ 1 wmð Þc0I M ð10cÞ

dc

dz¼ c0 1 cð Þ

2M zð Þ

1 c0I M þ R0S I S

ð10dÞ

C m f zd ð Þ ¼ C ch

f zd ð Þ ¼ C 0

f ¼ C 0s

km þ 1 kmð ÞF d ð10eÞ

F zd ð Þ ¼ F d and c zd ð Þ ¼ 0: ð10f ;gÞ

The four dimensionless parameters are

Da ¼ mh

rsV 0s

, ¼ h

rsV 0s

, R0S ¼ S 0h

rsV 0s

,c0 ¼ M 0h

rsV 0s

ð10hÞ

where Da is a melting Damko « hler number; and R0S arethe dimensionless channel^matrix chemical exchange rate

and melt suction rate, respectively; w0 is a reference orcharacteristic channel volume fraction, as explainedbelow; I M and I S are shorthand notations representing thedimensionless integrals of the dissolution rate and meltsuction rate, respectively,

I M zð Þ ¼

Z zzd

M xð Þdx ¼ z zd ð Þ

1þb

1 þ bð Þ ð10iÞ

I S zð Þ ¼

Z zzd

S xð Þdx ¼ z zd þ a

1 þ ð Þ z zd ð Þ

1þ: ð10jÞ

Equations (10a)^(10d) were obtained by eliminating themass fluxes of the melt and the solid (product of mass frac-tion and velocity) from the 1-D steady-state version of equations (1c), (7a), (7b), and (8d). Equations (10e)^(10g)


133


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are the boundary conditions at the base of the two-porosity

column where the melt compositions in the channel andthe matrix are the same and given by the batch meltingequation [equation (10e)]. A derivation of equations(10a)^(10g) is given in Appendix B.

Channel volume fraction in an upwellingmelting columnBefore discussing the chemical evolution of thedouble-lithology system it is necessary to know how thevolume fraction of the high-porosity channels varies verti-cally in an upwelling steady-state mantle column.Variations in channel volume fraction in an upwellingtwo-porosity double lithology melting column have not

been considered in previous double porosity models (e.g.Iwamori, 1993, 1994; Lundstrom, 2000; Ozawa, 2001; Jullet al ., 2002; Liang, 2008; Liang & Peng, 2010). Equation(10d) states that vertical variations in channel volume frac-tion arise when the matrix^channel transformation rate isnon-zero. The channel volume fraction can be determinedonce the dissolution rate and melt suction rate are given.

When the matrix^channel transformation rate and meltsuction rate are constant and uniform (b¼a¼0), the

channel volume fraction can be solved exactly from equa-

tion (10d),

c ¼ 1 1

1 c0c0þR

0S

ln 1 c0 þ R0S

z

: ð11aÞWhen the net dissolution and melt suction rate is small(c0 þ R

0S


11/28

equation (10d) can be integrated numerically. Figure 2acompares the channel spatial distributions in a steady-stateupwelling column for three choices of the matrix^channeltransformation rates (continuous lines labeled w1, w2,and w3; see figure caption for parameters used in thecalculations). As expected, the channel volume fraction

in the upwelling column depends strongly on the matrix^channel transformation rate. The effect of melt suctionon channel distribution is also explored in Fig. 2a (dash^dotted curve). For the relatively small melt suctionrates relevant to melt migration beneath a mid-oceanridge (R0S Da < 0 25), the channel volume fractionis practically independent of the melt suction rate.Neglecting melt suction, equation (10d) can be integratedexactly,

c ¼ 1 1

1 ln 1 c0z

1þb

1 þ bð Þ

: ð12Þ

Channel melt composition and itsdependence on melt suction andchannel formationVariations in channel melt composition are due to flow of melt from the lower part of the channel to the overlyingchannel (advection), vertical mixing of melts extractedfrom the matrix (melt suction) with the melts advectedfrom the channels below, matrix dissolution, and diffusiveexchange between the melt in the channel and the melt inthe matrix. In the absence of diffusive exchange, weexpect the channel melt at a given position to be a mixture

of matrix melts extracted from various depth in the upwel-ling column, hence a weighted average of the matrix melt.This can be shown by considering a reduced problem inwhich R0S > 0 (i.e. melt is extracted from the matrix intothe channel) and ¼ 0 (i.e. no diffusive exchange betweenthe melt in the channel and that in the matrix). Equation(10b) then reduces to

dC ch f dz

¼C m f C

ch f

R0S S zð Þ þ c0 k

E mC

m f k

E chC

ch f

M zð Þ

R0S I S þ c0kE chI M

ð13Þ

where the dimensionless melt suction and matrix^

channel transformation functions are all positive[S (z)> 0, M (z)> 0]. Recalling the definitions of thedimensionless integrals, I S and I M , equation (13) can berearranged

d R0S I S þ c0kE chI M

C ch f

n odz

¼ R0S S zð Þ þ c0kE mM zð Þ

C m f

ð14Þ

which can be integrated for the channel melt composition

C ch f zð Þ ¼

R zzd

R0S S xð Þ þ c0kE mM xð Þ

C m f xð Þdx

R zzd

R0S S xð Þ þ c0kE chM xð Þ

dx

: ð15Þ

The term R0S S C

m f in equation (15) is the mass flux of achemical species extracted from the matrix into the chan-

nel, while the term c0kE mM C

m f is the mass flux of a chemi-

cal species in the bulk matrix converted into the channel(sold þ melt). The integrand in the numerator of equation(15) is the accumulated contribution of the total mass fluxof a chemical species at a give point in the matrix to thechannel. Hence, in the absence of channel^matrix diffusiveexchange, the composition of the channel melt is simply aweighted average of the melt extracted to the channel andthe amount of matrix materials transformed into channel.The channel melt is therefore the ‘pooled’ or ‘aggregated’matrix melt. This point can be further illustrated by con-sidering a simple case.

When the matrix^channel transformation rate and themelt suction rate are constant and uniform (S ¼ M ¼1),the channel melt composition is proportional to the aver-age matrix melt composition, namely,

C ch f zð Þ ¼ R0S þ c0k

E m

R0S þ c0kE ch

z zd ð Þ

Z zzd

C m f xð Þdx: ð16Þ

When R0S c0kE m wm þ 1 wmð Þkm½ c0 and R

0S

c0kE ch wch þ 1 wchð Þkch½ c0, which can be realized

when the trace element of interest is highly incompatibleand the mass fractions of the melt in the matrix (wm) andthe channel (wch) are small or/and when the volume frac-

tion of the channel is very small, the channel melt compo-sition is simply the vertical average of the matrix meltcomposition, namely,

C ch f zð Þ ¼ 1

z zd

Z zzd

C m f xð Þdx ¼ 1

F F d

Z F F d


The RHS expression of equation (17) has been widely usedto calculate the aggregated melt compositions during frac-tional or dynamic melting. The physical meaning of thissimple averaging can now be understood in terms of extraction of melt from the matrix into the channel, whenthe effect of matrix^channel transformation is negligibly

small. Equations (16) and (17) also show that the highlyincompatible or compatible trace element abundance inthe channel melt is not very sensitive to the melt suctionrate so long as R0S c0k

E m and R

0S c0k

E ch or R

0S c0k

E m

and R0S c0kE ch. This point will be further illustrated by

the example given in Case (c) below.The role of matrix^channel transformation in affecting

channel melt composition can also be assessed with thehelp of equation (15) or (16). Recalling the definition of the


135


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effective matrix/melt and channel/melt partition coeffi-cients [equation (8c)], the products, kE mC

m f and k

E chC

ch f , are

the abundances of a chemical species in the bulk matrixand channel (solid þ melt), respectively. For highly incom-patible trace elements in the matrix and the channel, theeffective partition coefficients kE m and k

E ch are expected to

be small (1). Matrix^channel transformation thereforewill have a negligibly small effect on the channel melt com-position if the matrix dissolution rate is not significantlylarger than the melt suction rate (i.e. R0S S > c0k

E mM ). For

compatible trace elements in the matrix and the channel,the effective partition coefficients kE m and k

E ch are expected

to be large (>1). The effect of matrix dissolution and chan-nel formation must be included in calculating the channelmelt composition. When kE ch > k

E m > 1, which is the case

for Ni in a dunite channel and a harzburgite or lherzolitematrix, the compatible trace element abundance in thechannel melt is lower than that in the matrix melt at agiven position in the upwelling column. These general fea-tures of the channel melt for both the compatible andincompatible trace elements will be further elaborated bythe examples below.

Melt compositions in the matrix andthe channelGiven channel volume fraction, we can now calculate themelt compositions in the matrix and the channel and thedegree of melting experienced by the solid matrix. Tobetter understand the variations in the channel andmatrix melt compositions and their response to the varioussource or sink terms in equations (10a)^(10c), we considerfour simple cases below: (a) constant matrix^channeltransformation rate and melt suction rate; (b) variable

melt suction rate; (c) variable matrix^channel transforma-tion rate; (d) diffusive and dispersive exchange betweenthe melt in the channel and that in the matrix

Case (a): constant matrix^channel transformation rateand melt suction rateIwamori (1994), Ozawa (2001), and Liang & Peng (2009)gave analytical solutions for a trace element in the matrixmelt for cases of constant channel volume fraction(M cm ¼ 0) and constant or linearly variable melt suctionrate. Here we consider a more general problem in which thematrix^channel transformation rate and melt suction rateare constant and uniform in the double lithology region.

Assuming M (z) ¼ S (z) ¼ 1 and ¼ 0, equations (10a)^(10c) can be solved exactly. The analytical solutions are

C m f ¼ C 0s

km þ 1 kmð ÞF d

1 þ1 km S kE m

Da z zd ð Þ

km þ 1 kmð ÞF d

km 11 km S k

E m

ð18aÞ

C ch f ¼ C 0s

SþkE ch

Da zzd ð Þ

1 1þ1km SkE m

Da zzd ð Þ

km þ 1kmð ÞF d

SkE m1km Sk

E m

8>><>>:

9>>=>>;

ð18bÞ

1 F

1 F d ¼ 1

1 þ 1 wmð Þ½ Da z zd ð Þ

1 F d ð Þ

11 þ 1 wmð Þ

ð18cÞ

where S and H are the dimensionless melt suction rate andmatrix^channel transformation rate, defined with respectto the scaled matrix melting rate, respectively,

S ¼ S 0

0m

¼ S 0

m 1 cð Þ, ¼

M 0

0m

¼ M 0

m 1 cð Þ: ð18dÞ

The channel volume fraction increases almost linearly as afunction of z in the upwelling double lithology column(Fig. 2a) and can be calculated using the expression givenby equation (11a) or (11b). The degree of melting experi-enced by the solid matrix also increases approximately lin-early as a function of z in the double lithology columnwhen the dimensionless melting rate Da is small (bluedashed line F 1 in Fig. 2a). This can be seen by expandingequation (18c) in a Taylor series,

F zð Þ F d þ Da z zd ð Þ 1 þ 1 wmð Þ

2 1 F d ð Þ Da z zd ð Þ

:

ð18eÞ

Hence F also increases with the increase of matrix^channeltransformation rate (green and red dashed lines F 2 andF 3 in Fig. 2a). As more matrix converts into channel, themass flux of the solid matrix decreases [equation (B4d) inAppendix B]. Consequently the degree of melting experi-enced by the solid matrix also increases [equation (10c)].The effect of increasing matrix^channel transformationrate on the fractionation of the trace element abundancesin the matrix melt is discussed in Case (c) below.

Equation (18a) reduces to the double porosity formula-tion of Iwamori (1994) when Da is small and H¼ F d ¼ 0or a simplified formulation of Ozawa (2001, his equation68) when Da is small and H¼ 0. When Da is small,S

þ k

E

m

¼ 1, and F d ¼ 0, equation (18a) reduces to theperfect fractional melting model for the matrix melt. Itshould be noted that this condition is specific to the givenelement of interest. Further, equation (18b) is differentfrom the expression for the aggregated (perfect) fractionalmelt, by a factor of 1= S þ kE ch

(equation 16), as a result

of matrix dissolution and channel formation. Nevertheless,this difference is very small for highly to moderatelyincompatible trace elements during near fractional melting


136


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as S þ kE m ! 1. When F d > 0, we can determine theupper and lower bounds for S by examining the relativematrix melt flux,

!m f fmV

m f

s 1 fm

V m

s

1 S wmð ÞF þ S þ wmð ÞF d 1 1 þ 1 wmð Þ½ F þ 1 wmð ÞF d

ð19Þ

where the approximate expression was obtained fromequations (B4c) and (B4d) in Appendix B. Because thevertical melt flux, melt suction rate, and matrix^channeltransformation rate are all non-negative, we have0 S þ wmð Þ F = F F d ð Þ. Hence the upper boundfor the dimensionless melt suction rate is a function of z inthe melting column,

Smax zð Þ ¼ F = F F d ð Þ wm: ð20Þ

In the presence of a single lithology lower region,Smax > 1, as the melt fraction in the matrix is in generalvery small. The case of Smax zð Þ ¼ Smax, therefore, may betaken as the limit of perfect fractional melting at a givenposition z in the double lithology column, as all the meltproduced in the matrix is segregated into the channel at z(Liang & Peng, 2010).

Figure 3a^c compares the melt compositions in thematrix (continuous lines) and the channel (dashed lines)

calculated using equations (18a)^(18c) and H¼1 (bluelines) with the melt compositions in the matrix and thechannel calculated using equations (18a)^(18c) and H ¼ 0(red lines) for three trace elements [km ¼ (0·01, 0·12, 7·5),kch ¼ (0·0001, 0·01, 10)] and a normalized melt suction rate

S¼ 0·95. The melt composition in the single lithology

region (0< z < 0·136) is calculated using the batch meltingmodel. The channel volume fraction increases linearlyfrom 0 to 22% in the double lithology column(0·136< z


14/28

batch melting model for the matrix melt. This can beunderstood in terms of flow and mixing between the lessdepleted small-degree melt in the lower part of the channeland the more depleted large-degree melt in the upperpart of the melting column. As shown in Fig. 3a and b,the abundance of an incompatible trace element in the

channel melt is dominated by the less depletedsmall-degree melts extracted in the lower part of the melt-ing column. Further, dissolution of pyroxene in the lessdepleted matrix and precipitation of olivine in the channelinthe lower part of the melting column increases the moder-ately incompatible trace element abundance in the channelmelt relative to the case of H¼ 0 (compare the blue andred dashed lines in Fig. 3b) because the trace element of interest is still highly incompatible with respect to the solid(e.g., olivine) in the channel (km ¼ 0·12 and kch ¼ 0·001).

Finally, consistent with our analysis in the previous sec-tion, matrix dissolution dominates the compatible traceelement abundances in the channel melt (blue dashed line

in Fig. 3c). Because the trace element of interest is morecompatible in the channel (kch ¼10) than in the matrix(km ¼ 7·5), pyroxene dissolution from the matrix and oli-vine precipitation in the channel lower the compatibletrace element abundances in the channel melt significantly.[The abrupt decrease in the compatible trace elementabundance at z ¼ zd (¼ 0·136) is probably due to our use of a (constant) matrix^channel transformation rate thatis independent of the channel^matrix interfacial area.]

This is in sharp contrast to the double porosity models of Iwamori (1994), Ozawa (2001), and Liang & Peng (2009)in which the effect of matrix dissolution is not included(red dashed line in Fig. 3c). The channel melt derivedfrom the aforementioned models is simply a mixture of melts extracted at various depths along the melting

column. Hence compatible trace element abundances inthe channel melt are most sensitive to matrix dissolution.This point is further illustrated in Case (c) below.

The depth of channel initiation (zd ) and the magnitudeof the relative melt suction rate are important in determin-ing the extent of fractionation of the incompatible traceelements in an upwelling melting column. Figure 4a^c dis-plays the melt compositions in the matrix (continuouslines) and the channel (dashed lines) for three trace ele-ments [km ¼ (0·01, 0·12, 7·5), kch ¼ (0·0001, 0·01, 10)] calcu-lated using equations (18a)^(18c) (H¼1 and ¼ 0) forthree choices of the relative melt suction rates (S¼ [0·25,0·95,1·25]) and a slightly shallower depth of dunite channel

initiation than that in Fig. 3a^c (zd ¼ 0·

21 and F d ¼ 5%).The upper bound for the relative melt suction rate at thetop of the melting column is Smax ¼1·253 in this case. Asshown in Fig. 4a and b, the matrix melt compositions forthe case of S ¼ 0·25 (continuous red lines) are very closeto that for batch melting. The strong dependence of theincompatible trace element abundances in the matrix melton the melt suction rate was noted previously (Iwamori,1994; Ozawa, 2001; Liang & Peng, 2009) and can be easily

10−1

100

101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cmf /C

0s

z

100

101

Cmf /C

0s

0.1 0.12 0.14 0.16 0.180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cmf /C

0s

S = 0.95

S = 1.25

S = 0.25

Batch

(a) (b) (c)

c h

a n n e

l

m a t r i x

c h a n n e

l

m a t r i x

c h a n n e l , k

c h

= 1 0

- 5

m a t r i x , k

m = 0 . 0 1

km = 0.12

kch = 0.001

km = 10

kch = 7.5

Fig. 4. Variations of three trace elements in the matrix melt ( km ¼ 0·01, 0

·12, and 10

·0, continuous lines) and the channel melt (kch ¼ 0

·00001,

0·001, and 7·5, dashed lines) in a steady-state upwelling melting column for three choices of the normalized melt suction rate [ S ¼ (0·25, 0·95,1·25)]. The continuous lines and dashed lines represent melt compositions calculated using equations (18a)^(18c) and for the case of a constantmatrix^channel transformation rate (H¼1). The channel volume fraction increases linearly from zero to 19% in the double lithology column(0·21


15/28

understood in terms of vertical melt flow in the matrix.Advection brings the less depleted melt produced by asmaller degree of melting in the deeper part of the meltingcolumn to the overlying melting region, increasing theincompatible trace element abundance in the melt and thesolid in the matrix. The extent of such self-enrichmentdepends on the mass flux of the melt relative to that of thesolid, !m (and hence S). As more less-depleted melt is flow-

ing from the lower part of the melting column into theoverlying matrix (!m increases and S decreases), theincompatible trace element abundance in the matrix meltincreases. This melt-flow induced self-enrichment in thematrix melt also explains the zd dependence of the incom-patible trace element abundance in the matrix melt (com-pare the blue continuous lines between Figs 3a and 4a,and Figs 3b and 4b, respectively): the shallower thedunite channel initiation (the larger the F d ), the larger themass flux of the melt at the base of the dunite channel[!m(zd ) ¼ F d /(1 ^ F d )] is, and, therefore, the higher theabundance of the incompatible trace abundance in theoverlying double lithology column. This self-enrichment is

especially significant for the highly incompatible trace ele-ments, but barely noticeable for the compatible trace ele-ments (compare Figs 3c and 4c), as the abundances of thelatter are buffered by the solid.

Case (b): spatially variable melt suction rateHere we further explore the effects of melt suction on theabundance and distribution of the trace elements in thematrix melt and the channel melt by examining the case

of spatially variable melt suction rates. Figure 5a^c com-pares the trace element abundances [km ¼ (0·01, 0·12, 7·5),kch ¼ (0·0001, 0·01, 10)] in the matrix melt (continuouslines) and the channel melt (dashed l ines) for three choicesof melt suction rate: (a) a constant and uniform melt suc-tion rate (S¼0·75); (b) the melt suction rate decreases lin-early from the bottom of the double lithology column(S ¼0·96) to the top (S¼0·54); (c) the melt suction rate

increases as a function of z in the double lithology column(from 0·54 to 0·96). The average melt suction rates for thelatter two cases are the same as in case (a) (Fig. 2b). Themelt compositions, degree of melting, and channel volumefraction were calculated by numerical integration of equa-tions (10a)^(10d) with F d ¼ 0·03, H¼1, and ¼ 0. Asshown in Fig. 5a^c, a spatially variable melt suction ratehas little or no effect on the compatible and incompatibletrace element abundances in the channel melt as well asthe compatible trace element abundance in the matrixmelt. However, the incompatible, especially the highlyincompatible, trace element abundances in the matrixmelt (and solid) are very sensitive to how the melt suction

rate varies spatially in the double lithology column (Fig.5a and b). In the case of strong melt suction at the bottomof the double lithology column (green curves in Fig. 5aand b), the solid matrix experiences near-fractional melt-ing in the lower part of the melting column. Consequentlythe incompatible trace element abundance in the residualsolid matrix is progressively more depleted towards thetop of the melting column. The opposite is true for thecase of weak to moderate melt suction near the base of

10−1

100

101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cmf /C

0s

z

100

101

Cmf /C

0s

0.1 0.12 0.14 0.16 0.180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cmf /C

0s

Batch

S1

S2

S3

(a) (b) (c)

c h

a n n

e l

m a t r i x

c h a n n e l

m a t r i x

c h a n n e l , k

c h

= 1 0

- 5

m a t r i x ,

k m = 0 . 0 1

km = 0.12

kch = 0.001

km = 10

kch = 7.5

Fig. 5. Effects of melt suction rate on the highly incompatible (a), moderately incompatible (b), and highly compatible (c) trace element abun-dances in the matrix melt (continuous lines) and the channel melt (dashed lines) in an upwelling steady-state melting column. Variations inthe melt suction rate are as shown in Fig. 2b.The melt compositions, degree of melting, and channel volume fraction were calculated by numer-ical integration of equations (10a)^(10d) with F d ¼ 0·03, H¼1 (w1 in Fig. 2a), and ¼ 0. Partition coefficients used in the calculations are ( km,kch) ¼ (0·01, 0·00001), (0·12,0·001), and (7·5,10) for (a ), (b), and (c), respectively.


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the double lithology column (red curves in Figs. 5a and5b). Interestingly, the incompatible trace element abun-dances and distributions in the matrix melt for the twocases considered are not symmetric about the referencecase of a constant melt suction rate (blue curves in Fig. 5a

and b). Hence incompatible trace element abundances inresidual peridotites may be used to infer spatial variationsin the relative melt suction rate during melting and meltmigration in the mantle (Liang & Peng, 2010).

Another interesting example of a spatially variable meltsuction rate is the case in which melt flows from the chan-nel back into the matrix in part of the melting column(i.e. a negative melt suction rate). This situation mayoccur in a wide dunite channel near the top of the meltingcolumn where compaction in the center of the channeldrives part of the channel melt into the surrounding harz-burgite (Schiemenz, 2009). Figure 6a and b displays twoincompatible trace element abundances [km ¼ (0·01, 0·12),kch

¼ (0·0001, 0·001)] in the matrix melt (continuous lines)and the channel melt (dashed lines) as a function of z fora case in which the dimensionless melt suction ratedecreases linearly from 0·95 at the bottom of the doublelithology column (zd ¼ 0·136) to ^0·90 at the top of themelting column (S ¼0·95[1 ^ 2·25(z ^ zd )], red lines). Themelt suction rate switches its sign at z ¼ 0·58. For reference,the case of near fractional melting with a constant meltsuction rate (S ¼0·95) is also shown in Fig. 6a and b (blue

lines). The most striking feature of the results shown inFig. 6a and b is the relative enrichment of the incompatibletrace elements in the matrix and the channel in the upperpart of the melting column where melt flows from thechannel into the matrix (z> 0·58). The highly incompati-ble trace element abundance in the matrix melt at z ¼1,in this case, is slightly higher than that in the underlyingmatrix (Fig. 6a). Percolation of the less depleted channelmelt into the more depleted matrix in the upper part of the melting column elevates the incompatible trace ele-ment abundances in the residual matrix. This effect ismore pronounced for highly incompatible trace elementsthan moderately incompatible ones (compare Fig. 6a andb), as the former are more sensitive to perturbations inmelt composition. As will be discussed further in the nextsection, scenarios similar to that outlined here may providea viable mechanism for shallow-level melt refertilizationor mantle metasomatism.

Case (c): spatially variable matrix^channel transformation rateTo further explore the effect of matrix dissolution andchannel formation on the abundance and distribution of trace elements in an upwelling steady-state meltingcolumn, we consider a more general case in which thematrix^channel transformation rate varies spatially(M cm ¼ M 0M (z)). Figure 7a^c compares the trace element

10−1

100

101

102

0

0.2

0.4

0.6

0.8

1

Cm

f /C

0

s

z

100

101

0

0.2

0.4

0.6

0.8

1

Cm

f /C

0

s

z

Batch

(a) (b)

S = S z( )

S = 0.95

c h a n n e l

m a t r i x

, l e

n n

a h

c

k c

h

0 1

=

- 5

m a t r i x ,

k m =

0 . 0 1

km = 0.12kch = 0.001

Fig. 6. Effects of a negative melt suction rate on the highly incompatible (a) and moderately incompatible (b) trace element abundances in thematrix melt (red continuous lines) and the channel melt (red dashed lines) in an upwelling steady-state melting column.The melt compositions,degree of melting, and channel volume fraction were calculated by numerical integration of equations (10a)^(10d) with S ¼ 0·95[1 ^ 2·25(z ^zd)], zd ¼ 0·136, H¼1 (w1 in Fig. 2a), and ¼ 0.The dimensionless melt suction rate decreases from 0·9 at zd ¼ 0·136 to ^0·42 at z ¼1. For refer-ence, the case of a constant melt suction rate (S¼0·95) is shown as blue lines. Partition coefficients used in the calculations are ( km,kch) ¼ (0·01, 0·00001) and (0·12,0·001) for (a) and (b), respectively.


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abundances [km ¼ (0·01, 0·12, 7·5), k ch ¼ (0·0001, 0·001, 10)]in the matrix melt (continuous lines) and the channelmelt (dashed lines) for two choices of matrix^channeltransformation rate (green and red curves w2 and w3 inFig. 2a) with the reference case of a constant dissolutionrate (blue line w1 in Fig. 2a). The melt compositions,degree of melting, and channel volume fraction were calcu-

lated by numerical integration of equations (10a)^(10d)with F d ¼ 0·03, S¼0·75, and ¼ 0. In spite of the largedifferences in w (by a factor of more than three at the topof the melting column, Fig. 2a), the incompatible trace ele-ment abundances in the matrix melt are rather similaramong the three cases compared, especially in the lowerhalf of the melting column (Fig. 7a and b). Further, theincompatible trace element abundance in the channelmelt is not very sensitive to channel distribution, especiallynear the top of the melting column.

The small but systematic decrease in the incompatibletrace element abundances in the matrix melt (Fig. 7a andb) and the increase in the compatible trace element abun-

dance in the matrix melt (Fig. 7c) at a given z

or F

as afunction of increasing matrix^channel transformation rateor channel volume fraction can be understood in terms of an increasing degree of melting experienced by the solidmatrix with the decrease of solid upwelling rate [comparedashed lines in Fig. 2a; see also equation (10c)].

The compatible trace element abundances in thechannel melt, on the other hand, depend strongly onthe matrix^channel transformation rate and spatial

distribution of the channel (Fig. 7c). For a small matrix^channel transformation rate (green dashed line in Fig. 7c,corresponding to w2 in Fig. 2a), the effect of melt suctionis not negligible compared with matrix dissolution, espe-cially in the lower part of the double lithology columnwhere the channel volume fraction is small (z ¼ 0·136 to0·7 in Fig. 7c). Consequently the compatible trace element

abundance in the channel melt decreases gradually in thedouble lithology region (between z ¼ 0·136 and 0·7), as anet result of channel formation (via pyroxene dissolutionand olivine precipitation), melt suction, and melt mixingwithin the channel. For a large matrix^channel transfor-mation rate (red and blue dashed lines in Fig. 7c, corre-sponding to w1 and w3 in Fig. 2a), the effect of meltsuction is small compared with matrix dissolution through-out the double lithology region for the compatible traceelement of interest. The compatible trace element abun-dance in the channel melt is not very sensitive to matrixdissolution and channel distribution (compare the red andblue dashed lines in Fig. 7c) and can be estimated usingthe simple expression

C ch f zð Þ km

kch z zd ð Þ

Z zzd


For the case considered in Fig. 7c where km/kch ¼ 0·75,the highly compatible trace element abundance in thechannel melt is about 25% lower than that in the averagematrix melt.

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100

101

102

0

0.1

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0.3

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0.5

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0.7

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0.9

1

Cmf /C

0s

z

100

101

Cmf /C

0s

0.1 0.12 0.14 0.16 0.180

0.1

0.2

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0.4

0.5

0.6

0.7

0.8

0.9

1

Cmf /C

0s

ψ 1

ψ 2

ψ 3

Batch

(a) (b)

c h

a n n e

l

m a t r i x

c h a

n n e l

m

a t r i x

(c)

c h a n n e l ,

k c h

= 1 0

- 5

m a t r i x

, k m = 0 . 0 1

km = 0.12kch = 0.001

km = 10kch = 7.5

Fig. 7. Effects of variable channel volume fraction on the highly incompatible (a), moderately incompatible (b), and highly compatible (c) traceelement abundances in the matrix melt (continuous lines) and the channel melt (dashed lines) in an upwelling steady-state melting column.Variations in the volume fraction of the channel are shown in Fig. 2a. The melt compositions, degree of melting, and channel volume fractionwere calculated by numerical integration of equations (10a)^(10d) with F d ¼ 0·03, S ¼0·75, and ¼ 0. Partition coefficients used in the calcula-tions are (km, kch) ¼ (0·01, 0·00001), (0·12, 0·001), and (10,7·5) for (a), (b), and (c), respectively.


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Case (d): diffusive and dispersive exchangeDiffusive and dispersive exchange between the melt in thechannel and that in the matrix contributes to the develop-ment of concentration boundary layers in the channel^matrix interfacial region. The effects of diffusive and

dispersive exchange in the melts on the abundance and dis-tribution of trace elements in the matrix melt and thechannel melt are explored in Fig. 8a^c for the case of aconstant matrix^channel transformation rate and a linearmelt suction rate (w1 in Fig. 2a and S2 in Fig. 2b). Forpurpose of illustration, we choose a dimensionlessexchange rate ¼ 2·5Da (red continuous and dashedlines in Fig. 8a^c). For comparison, a case of no diffusiveexchange is also shown (blue lines). For the exchange rateused in this example, diffusive and dispersive exchangebetween the channel melt and the matrix melt affectmostly the incompatible trace element abundances in thechannel and the matrix melt. Because the compatibletrace element abundances are buffered by the solid, verylittle change in the compatible trace abundances in thechannel and the matrix melt is observed. We should pointout that the abundances of the trace element in the channeland matrix melts shown in Fig. 8a^c are averaged valueswithin a REV in the channel continuum and the matrixcontinuum. Spatial compositional variations across thechannel^matrix interface cannot be captured by thepresent model.

In summary, the incompatible trace element abundancesin the matrix melt and solid are very sensitive to thedepth of high-porosity melt channel initiation, and thesign, magnitude, and variation of the relative melt suctionrate, and depend moderately to weakly on matrix^channeltransformation rate or channel volume fraction in thedouble lithology region. The incompatible trace elementabundance in the channel melt is dominated by the lessdepleted small-degree melts from the lower part of themelting column and not very sensitive to the details of how the melt suction rate and matrix^channel transforma-tion rate vary spatially in the double lithology region. Thecompatible trace element abundances in the melt and thesolid in the channel and the matrix depend strongly onmatrix dissolution and channel distribution. In general,the compatible trace element abundances in the channelmelt are lower than those in the average matrix melt, by afactor of up to km/kch. Diffusive and dispersive exchange inthe melt affect mostly the (highly) incompatible trace ele-

ment abundances in the channel and the matrix melt.

D I S C U S S I O N

Non-modal melting in the matrixcontinuumFor simplicity, we have neglected the effect of non-modalmelting in the model derivation. It is well known that

10−1

100

101

102

0

0.1

0.2

0.3

0.4

0.5

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0.7

0.8

0.9

1

Cmf /C

0s

z

100

101

Cmf /C

0s

0.1 0.12 0.14 0.16 0.180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cmf /C

0s

Λ = 2.5Da

Λ = 0

Batch

(a) (b) (c)

c h

a n n e l

m a t r i x

c

h a n n e l

m a t r i x

c h a n n e l , k

c

h = 1 0

- 5

m a t r i x

, k m =

0 . 0 1

km = 0.12kch = 0.001

km = 10kch = 7.5

Fig. 8. Effects of diffusive and dispersive exchange on the highly incompatible (a), moderately incompatible (b), and highly compatible (c) traceelement abundances in the matrix melt (continuous lines) and the channel melt (dashed lines) in an upwelling steady-state melting column.The case of ¼ 2·5Da [red continuous and dashed lines in (a^c)] and a reference case of no diffusive exchange ( ¼0) are shown as red andblue lines, respectively. The melt compositions, degree of melting, and channel volume fraction were calculated by numerical integration of equations (10a)^(10d) with Da ¼ 0·22, F d ¼ 0·03, H ¼1 (line labeled w1 in Fig. 2a), and ¼ 0. The melt suction rate used in the two cases arethe same and correspond to the line S2 in Fig. 2b. Partition coefficients used in the calculations are (km, kch) ¼ (0·01, 0·00001), (0·12, 0·001), and(10,7·5) for (a), (b), and (c), respectively.


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mantle melting is non-modal. For example, diopside,orthopyroxene, and spinel are preferentially consumedwhile olivine is precipitated during partial melting of aspinel lherzolite under normal mantle conditions (e.g.Kinzler & Grove, 1992; Baker & Stolper, 1994). Duringnon-modal melting, the bulk solid^melt partition coeffi-

cient km is related to the degree of melting experienced bythe solid matrix through the expression (e.g. Shaw, 1970)

km ¼k0m k

pmF

1 F ð22Þ

where k0m is the bulk solid-melt partition coefficient at theonset of melting; and k pm is the bulk solid^melt partitioncoefficient calculated according to the proportion of min-erals that participated in the melting reaction. Given therelationship among the bulk partition coefficients, we canobtain a conservation equation for a trace element in thematrix melt during non-modal melting. Substitutingequations (3a), (3b), and (8d) into equation (7b), we have

. . .f g @C m f

@t þ . . .f gr C m f ¼ . . .f g

þ 1 cð Þ k pm 1 þ F dk pmdF

C m f m

ð23Þ

where terms in {. . .} on the LHS of equation (23) are iden-tical to the corresponding terms given by equation (8b),whereas terms in {. . .} on the RHS of equation (23) arethe same as the first and the third terms on the RHS of equation (8b). Hence only the melting terms are differentbetween the modal and non-modal melting models[i.e. the last term on the RHS of equations (8b) and (23)],as the mineral proportions participating in the melting

reaction are different between the two cases. The deriva-tive dk pm=dF in equation (23) accounts for variations inmineral^melt partition coefficients during partial melting(for a detailed derivation, see Appendix A in Liang &Peng, 2010). As an example, we use equation (23) to calcu-late REE abundances in diopside in residual peridotite fortwo choices of melt suction rate below.

Melt suction rates as inferred fromincompatible trace element abundancesin peridotitesThe abundances of highly incompatible trace elements inthe matrix melt and solid are very sensitive to the degree

of melting experienced by the solid matrix (F ) and the rel-ative rate of melt suction (S) but insensitive to spatial vari-ations in channel volume fraction or matrix dissolution.Hence it may be possible to infer F and S from the incom-patible trace element abundances in residual peridotiteswithout explicit knowledge of the distribution of high-porosity melt channels or hydro-fractures in themantle. Iwamori (1

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J. Petrology 2010 Liang 125 52