Geochemistry, Geophysics, Geosystems

Melt extraction from partially molten peridotites



[1] We measured the rate of melt extraction from partially molten olivine aggregates into a porous reservoir as a function of melt content (ϕ), melt viscosity, grain size, and pressure difference. Samples were prepared using mixtures of olivine and either lithium silicate, MORB glass, or albite glass to obtain a range in melt viscosities. Experiments were conducted at 1473 K with a confining pressure (Pc) of 300 MPa. The melt pressure (Pm) was controlled using a pore fluid actuator to adjust the pressure of Ar gas (Pf) in contact with the melt. Compaction rates decrease with decreasing ϕ and increase linearly with the pressure difference ΔP = PcPf. For the two lower viscosity melts, no gradients in melt fraction were observed in samples with melt fractions as low as 0.03. By contrast, the melt fraction increases with distance away from the reservoir for samples with the highest viscosity albitic melt. These observations indicate that the compaction length for the lower viscosity melt experiments (i.e., olivine + MORB and olivine + Li-silicate) was greater than the sample length. However, the compaction rates of these samples were limited by two different processes. Melt transport is relatively easy in the coarse-grained Li-silicate samples where melt viscosity is lower. In this case, melt extraction is limited by viscous compaction of the solid matrix. In the olivine + MORB samples, the fine grain size and higher melt viscosity inhibits melt transport and limits extraction rates. The olivine + MORB data constrain the permeability of the partially molten samples. Permeability is proportional to ϕ3 for melt fractions from 0.05 to 0.35. In addition, assuming that permeability scales with the square of the grain size, the magnitude of the permeability compares very well with that measured for Fontainebleau sandstone and fine-grained quartz aggregates with similar pore-space topology. Calculations of the compaction length using previously published continuum models agree with the estimates derived from the experiments.

1. Introduction

[2] Our understanding of the evolution and dynamics of the Earth is strongly influenced by geochemical analyses of basalts and remote geophysical observations. The interpretation of the geochemical data depends critically on the influence of melt migration on the composition of the basalt [Kelemen et al., 1997; Kinzler and Grove, 1992; Maaløe and Scheie, 1982; McKenzie, 1984; Spiegelman, 1993]. Melt migration, in turn, is directly influenced by the permeability of the partially molten peridotite and the way permeability changes with deformation. On a larger scale, melt extraction plays an integral role on the dynamics of upwelling regions of convection cells beneath oceanic ridges, plumes, and subduction zones.

[3] Models of melt segregation [McKenzie, 1984; Ribe, 1987; Scott and Stevenson, 1989] rely on assumptions of mechanical and transport properties within a representative elemental volume (REV). Typically, the descriptions of these material properties are motivated by considerations of equal-sized grains with a melt topology controlled by surface energy [Cheadle, 1993; van Bargen and Waff, 1986]. The permeability of partially molten rocks has not been measured directly. However, it has been estimated based on the kinetics of melt redistribution in partially molten aggregates [Riley and Kohlstedt, 1991]. In addition, permeability has been measured for rocks that may be analogues for partially molten aggregates [Bernabé et al., 1982; Fischer and Paterson, 1992; Wark and Watson, 1998; Zhang et al., 1994].

[4] For melt to migrate out of a partially molten rock, the matrix must compact [Schwenn and Goetze, 1978]. Densification rates are significantly enhanced when melt is present [Cooper et al., 1989]. The rheology of partially molten peridotites has also been studied in triaxial compression tests at elevated confining pressure, but under undrained conditions, i.e., the melt content of the fully encapsulated charge remained constant during deformation [Beeman and Kohlstedt, 1993; Hirth and Kohlstedt, 1995a, 1995b; Kohlstedt et al., 1999]. These studies indicate that creep rates increase strongly with an increase of melt fraction.

[5] In this study, we measured compaction of partially molten olivine aggregates subjected to isostatic confining pressures, but with drained pore fluids (melt). The melt pressure in the aggregate was controlled by connecting the sample to a porous reservoir drained to an elevated gas pressure. Melt is expelled from the olivine matrix into the pressurized reservoir. The experimental set-up allows continuous monitoring of densification of the aggregate and of the volume flux of melt extracted. The experiments are designed to investigate the evolution of permeability and its relation to the rates and mechanisms of melt extraction from partially molten peridotite. Thus, these experiments can be used to constrain the properties of the REV to be used in the larger scale calculations, where stress, temperature, and melt content vary as a function of position [McKenzie, 1984; Ribe, 1987; Scott and Stevenson, 1989].

2. Experimental Details

2.1. Apparatus

[6] All experiments were conducted in an internally heated gas-apparatus (Paterson Instruments) equipped with a pore pressure system. A large argon reservoir controls the confining pressure (Pc). The pressure (Pf) in a smaller, separate argon reservoir is controlled using a servo-controlled volumometer [Fischer and Paterson, 1989] to provide a boundary condition on melt pressure (Pm) in the sample. The two Argon reservoirs can be connected to ensure zero pressure difference between them. During compaction experiments when the reservoirs are separated, the pressure difference (ΔP = PcPf) is known within 0.1 MPa, based on calibrations of two electronic pressure gauges against a Heise gauge.

[7] External displacement transducers record the position of the axial actuator and the volumometer piston with a resolution of 0.3 μm. The axial force is measured with an internal load cell providing a stress accuracy of 0.5 MPa. Temperature was monitored directly above the sample using a Pt/Rh-Pt thermocouple during hot-press experiments, and a chromel-alumel thermocouple sheathed in inconel and welded into a standard high-pressure fitting during compaction experiments. Furnace calibrations indicate that the temperature varies less than 2 K along the length of the sample (10 mm). The accuracy of the thermocouples is about ±10 K.

2.2. Sample Preparation and Assemblies

[8] Samples were prepared by cold-pressing mixtures of crushed San Carlos olivine with either MORB glass, albite glass, or a 2:1 mixture of Li-metasilicate (Li2SiO3) and quartz powders into a Ni shell at 150 MPa. The olivine powder had an initial grain size of 15–30 μm, but that was further reduced during cold-pressing. Electron microprobe analyses indicate an average composition of ∼Fo90, consistent with the crystal density of 3337 kg/m3 as determined by the Archimedes method. The cold-pressed samples were then hot-pressed isostatically at 1523K and a confining pressure of 300 MPa for 5 hours. After the hot-press (HIP), a thin disc was cut from the densified aggregate for microstructural analysis. The remainder of each sample was then machined to form a cylinder 10 mm long with a diameter of ∼9.5 mm. Densities of the hot-pressed samples are reported in Table 1.

Table 1. Sample Properties After Hot Isostatic Pressing and After Compaction
Exp. #CompositionTime (min)MacroscopicaExperimental RecordbImage Analysisc
ρHIP,A (g/cm3)Δlmacro (mm)Δlax.disp. (mm)Δϕax.disp. (%)Δϕvol. (%)ϕ0 (%)ϕend(%)Δϕim (%)d′0 (μm)d′end (μm)
  • a

    ρHIP,A: density of hot isostatically pressed samples (Archimedes' method); Δlmacro: the difference in macroscopic length of samples before and after compaction.

  • b

    Δϕax.disp.: change in melt content calculated according to (1) and (3) using the total length change recorded by the external displacement transducer, Δlax.disp.

  • c

    Δϕim: change in melt content determined from the difference between initial (ϕ0) and final (ϕend) glass content; d′end: grain size of samples after compaction in two-dimensional sections described by the diameter of an area-equivalent circle.

  • d

    Limix = a mixture of 67.1wt.%Li2SiO3 + 32.9wt.%SiO2.

C-372ol + 30wt.%MORB1303.1700.750.7018.914.834.1 ± 4.816.4 ± 2.517.7 ± 5.4nd12.8
C-423ol + 18.3wt.% Limixd2503.0231.33 ± 0.20.9725.424.028.6 ± 3.49.1 ± 2.219.5 ± 4.015.820.2
C-424ol + 30wt.%albite3102.9850.800.4710.720.036.4 ± 3.428.6 ± 7.47.8 ±
C-425ol + 10wt.%MORB4603.2650.230.246.517.512.7 ± 0.55.8 ± 1.46.9 ± 1.511.013.4
C-450"2603.2960. ± 0.56.2 ± 1.46.5 ± 1.5nd13.7
C-479ol + 6.1wt.% Limix3703.2300.380.378.511.511.2 ± 1.63.0 ± 1.78.2 ± 2.319.424.9

[9] The assembly used in the melt extraction experiments is shown in Figure 1. The vitreous carbon spheres constituting the porous reservoir are 80–200 μm in diameter. A split alumina spacer was placed above the carbon reservoir to allow communication with the pore-fluid reservoir, but inhibit extrusion of the carbon spheres into the thermocouple bore. Samples were encapsulated in Ni sleeves to buffer the oxygen fugacity near Ni/NiO [Beeman and Kohlstedt, 1993].

Figure 1.

Sample assemblies. The samples are about 9.5 mm in diameter. During the hot-pressing step the sample is undrained, but during the compaction experiments the sample was drained into a reservoir of glassy carbon beads.

2.3. Procedure

[10] Melt extraction experiments were conducted at 1473 K and a confining pressure of 300 MPa. The pore fluid pressure was varied from 299 to 250 MPa, yielding pressure differences between 1 and 50 MPa. Each experiment involved the following steps: (1) compacting the carbon reservoir at a temperature below the solidus of the sample; (2) heating to the final temperature above the solidus of the sample; (3) increasing the pressure difference to initiate compaction of the sample and concomitant melt migration into the reservoir; and (4) quenching the sample.

[11] The initial compaction of the carbon reservoir was done at about 1230 K. During this step, the pressure difference was increased up to 100 MPa to over-compact the carbon reservoir while the sample was still solid. Calibration experiments in which the sample was replaced by a solid alumina cylinder showed that this procedure minimizes further compaction of the carbon reservoir at the smaller pressure differences, but higher temperatures used during melt extraction (Figure 2). Examination of the assemblies after calibration experiments demonstrated that there was no sintering or permanent deformation of the carbon spheres.

Figure 2.

Compaction behavior of carbon reservoir. In these tests the sample was replaced by a solid alumina spacer. The volumetric strain was measured using a pore pressure volumometer (dashed, smoother curves) and by tracking the bottom of the sample column (solid, jagged curves). Curves labeled 1–3 show the compaction of the reservoir at 1230 K for pressure differences of 50 MPa, 100 MPa, and the 50 MPa again. As a final step, compaction was monitored at 1473 K and 10 MPa pressure difference (curve 4). During all heating steps, the pressure difference was zero.

[12] We used two procedures for heating after the reservoir-compaction phase. During the first experiment (C-372; Figure 9), temperature was increased above the solidus while a small pressure difference was applied, resulting in some melt segregation before the temperature equilibrated. In subsequent experiments, the temperature was increased while the pore fluid and confining pressure systems were connected to ensure that there was no pressure difference on the sample. As shown by calibration tests on alumina samples, this second procedure resulted in some decompaction of the reservoir (Figure 2). However, the procedure had the advantage that the melt fraction at the beginning of melt extraction was well constrained.

[13] At run conditions, melt extraction was initiated by lowering the pore fluid pressure, resulting in a pressure difference on the sample. Some samples were subjected to intervals at several pressure differences; and, in a number of those cases, identical pressure differences were applied at different stages of an experiment. During each interval the compaction rate and the total volume of melt extracted could be measured at any given instant. Experiments were terminated when the carbon reservoir was filled with melt or when compaction rates approached those determined for the carbon reservoir alone. To terminate the experiment, we connected the pore fluid and confining pressure systems again (ensuring ΔP = 0) and then quenched the sample by shutting off power to the furnace. The average cooling rate between 1473 K and 1173 K was ∼150 K/min. Pressure was reduced after the temperature decreased well below the solidus.

2.4. Microstructure Analysis

[14] After a compaction experiment, the entire assembly was impregnated with epoxy before it was cut parallel to the cylindrical axis. Scanning electron (SEM) micrographs were taken from polished surfaces in both the hot-pressed and the post-extraction samples. Melt-fraction and distribution were determined by digital image analysis of the SEM micrographs to complement the volumetric strain data. Apparent dihedral angles were estimated from distributions of angles enclosed by opposing grains. Individual measurements have a reproducibility/operator bias of about ±2°. Olivine grain size was determined from perimeter and cord-length analyses. The compositions of the olivine matrix and the quenched glasses were analyzed using an electron microprobe.

2.5. Data Analysis

[15] Volumetric strain and strain rate were calculated using microstructure observations, changes in sample length recorded by the position of the axial actuator, and changes in reservoir volume recorded by the position of the volumometer actuator. A small constant end-load of about 1 to 2 MPa was kept on the samples so that a continuous record of changes in sample length could be obtained from the position of the axial actuator. Assuming isotropic compaction, the volumetric strain (εV) is calculated from the difference Δl between initial (lo) and current sample length (l) according to

equation image

where compaction is positive. The resolution in actuator travel yields a resolution in length change equivalent to 3 × 10−5 axial strain, which for isotropic compaction is a volumetric strain of ∼10−4.

[16] As melt is expelled from the sample into the carbon reservoir, the volumometer piston retracts to keep pore fluid pressure constant. Then, the displacement (b) of the volumometer piston with a cross-sectional area of Avol is related to the change in the connected melt volume in the sample (ΔVmelt) by the relationship: b · Avol = ρP,TP,293K · ΔVmelt. The factor ρP,TP,293k accounts for the difference in Argon density at the temperature conditions of the reservoir and that of the volumometer at given pore fluid pressure. Volumetric strain can then be calculated using the relationship

equation image

where V0 denotes the initial sample volume. At our experimental conditions, the resolution in piston travel yields a resolution equivalent to 2 × 10−5 volumetric strain.

[17] For the large compactions analyzed here, the volumetric strain is not equivalent to the change in melt content. Instead, the current melt content (ϕ) is related to the volumetric strain by

equation image

By evaluating (3) for the final melt content (ϕf), the microstructural analyses can be directly compared to the recorded volumetric strain; the change in melt fraction Δϕ = ϕ0 − ϕf = (1 − ϕ0)(exp εv − 1). Finally, the melt extraction rate equation image and equation image are related by equation image

3. Results

3.1. Melt Viscosity and Microstructures

[18] Melt viscosities were calculated using electron microprobe analyses of quenched melt pools within the samples and in the carbon reservoirs (Table 2). The composition of the melt within the olivine + MORB samples is deficient in Mg and Fe relative to that in the carbon reservoir, indicating some quench modification. However, applying olivine + melt geothermometry [Gaetani and Grove, 1998] to the melt expelled into the reservoir indicates the correct run temperature for Fo90. Back fractionation reveals that the two melt compositions are consistent. Within the Li-silicate samples the melt separates into two liquids upon quenching (Figure 3). The darker glass is almost a pure Li-silicate, while the brighter glass contains Mg and Fe from dissolved olivine. By contrast, melt expelled into the reservoir of these same samples appears homogeneous in SEM, and microprobe analyses agree with the phase diagram [Murthy and Hummel, 1955]. The glass phase in samples produced from olivine + albite gained an olivine component compared to stoichiometric albite. Viscosities for the melts at 1473 K (Table 2) were calculated using Shaw [1972]. The three systems span a range of four orders in viscosity with the albitic melt being most viscous.

Figure 3.

Backscattered electron images of the quenched partially molten olivine aggregates. The images on the left show the microstructure of samples after hot-isostatically pressing for 5 hours at 1523 K and 300 MPa. The images on the right show the microstructures of the same samples after melt extraction at 1473 K, a confining pressure of 300 MPa, and various pore pressures (Table 3). The scale bars apply to all but the last image. (a) Olivine + MORB before and (b) after compaction (C-372, final melt content ϕf = 16.4 ± 2.5). (c) Olivine + MORB before and (d) after compaction (C-450, final melt content ϕf = 6.2 ± 1.4). (e) Olivine + Li-silicate before (ϕ0 = 28.6 ± 3.4) and (f) after compaction (C-423, ϕf = 9.1 ± 2.2). (g) Olivine + Li-silicate before (ϕ0 = 11.2 ± 1.6) and (h) after compaction (C-479, final melt content ϕf = 3.0 ± 1.7). (i) Olivine + albite-glass before (ϕ0 = 36.4 ± 3.4) and (j) after compaction (C-424). The latter image was taken close to the interface between sample and carbon reservoir where pronounced compaction was observed; the melt content of the opposite sample end remained unaltered (see Figure 8). (k) Glass in carbon reservoir after quench of an olivine + MORB experiment (C-372).

Figure 3.


Figure 3.


Table 2. Melt Compositions and Calculated Viscosities
Flux glass compositiona (wt.%)Li-silicate in reservoirMORBalbite in pools
  • a

    Microprobe analysis; the composition of the Li-silicate glass was estimated from the analysis without Li2O.

  • b

    Estimated after Shaw [1972].

  in poolsin reservoir 
Al2O3 191618
CaO 1413 
Na2O 3210
TiO2 11 
melt viscosityb at 1473 K (Pa·s)213040104

[19] The microstructures of our samples are similar to those previously reported for partially molten olivine aggregates [Beeman and Kohlstedt, 1993; Hirth and Kohlstedt, 1995b]. Other than the change in melt content, microstructures are not noticeably altered by the melt extraction. After extraction of ∼20% melt, a sample originally containing ∼35vol.% MORB glass (Figure 3a) exhibits no obvious differences from a sample synthesized with an initial melt content of ∼12% (compare Figures 3b and 3c). There is no shape-preferred orientation of grains or melt pockets.

[20] There are systematic differences between the microstructures of samples with different melt compositions. Many olivine grains in contact with the MORB and the lithium silicate melt develop rational faces (Figures 3a3b, 3e3f). Only some of the small olivine grains in contact with the albitic melt show crystal faces (Figures 3g3h); the majority of grains exhibit gently curved boundaries. The largest grains in the albitic melt samples are odd-shaped with boundaries alternating between convex and concave, indicating that they originated from coalescence of several small grains rather than Ostwald ripening [Renner et al., 2002]. During hot isostatic pressing more coarsening of the olivine particles of the starting powder occurred in the lithium-silicate samples than in either the MORB or albitic melt samples (Figure 4).

Figure 4.

Histograms of olivine grain size in samples after melt extraction. The mean equivalent diameter is shown. (a) olivine + albite-glass (C-424) (b) olivine + MORB (C-425), and (c) olivine + Li-silicate (C-423).

[21] Analysis of the quenched samples demonstrates that all three solid-liquid systems have mean dihedral angles significantly below 60°. Thus, while not fully wetting, the liquid phase is interconnected along channels on three-grain edges. Cumulative histograms shown in Figure 5 indicate dihedral angles between 20 and 25°, with the average angle in the olivine + albite system being the smallest, while that in the olivine + Li-silicate system is the largest. Similar mean dihedral angles have been measured for the olivine + MORB system in previous studies [Riley and Kohlstedt, 1991; Waff and Bulau, 1979]. The asymmetry of the cumulative distributions may be associated with anisotropy in the interfacial energy [Jurewicz and Jurewicz, 1986].

Figure 5.

Cumulative frequency of dihedral angles in samples after melt extraction. About 150 individual measurements were done per sample on SEM images. The angle at a frequency of 0.5 corresponds to the apparent isotropic dihedral angle θ [Harker and Parker, 1945; Stickels and Hucke, 1964]. The low dihedral angle measured for all three systems indicates the melt phase is highly interconnected, i.e., θ < 60° The asymmetry of the distribution indicates some anisotropy in surface energy, in particular for olivine in contact with the Li-silicate melt.

[22] Comparisons of microstructures and grain size measurements before and after melt extraction demonstrate that a modest amount of grain growth occurred during compaction (Figure 3 and Table 1). The compaction phase of the experiments lasted between 2 and 8 hours. The microstructural observations agree with the finding that small amounts of melt, i.e., less than 5%, inhibit grain growth in olivine aggregates for these time intervals [Faul, 2000; Hirth and Kohlstedt, 1995a, 1995b]. The kinetics for grain growth in partially molten aggregates probably change systematically with increasing melt content. Most likely, this effect cannot be ignored for samples that compact by 20% during compaction. Thus, to estimate grain-size evolution during compaction one must account not only for the elapsed time, but also for changes in melt content. We used the final grain size in our analysis, because growth laws containing both parameters are lacking and growth was modest; we comment on the uncertainty this assumption introduces where necessary.

3.2. Strain Analysis

[23] The microstructural analyses, changes in macroscopic sample geometry, and the records of the external displacement transducers all indicate that the volumetric strain of the samples is isotropic and homogeneous for all but the olivine + albite-glass sample. As shown in Figure 6a, the length change recorded by displacement of the axial actuator agrees well with the macroscopic length change determined from measurements before and after the extraction experiments. An example of a compaction curve is shown in Figure 7 for an experiment with an initial melt content of 12% MORB. The volumetric strain determined from the volumometer and from the axial actuator displacements using (1) and (2) also agree. Finally, the change in melt content derived from the change in sample length and from image analysis of SEM micrographs of samples before and after melt extraction experiments also agree (Figure 6b).

Figure 6.

(a) Comparison between the length change recorded by the axial displacement transducer and the macroscopic length change. The error bar for the most deformed Li-silicate sample represents uncertainty arising from a convex end face observed for this sample. (b) Comparison between the change in melt content derived from the length change recorded by the axial displacement transducer (assuming isotropic compaction) and the change in melt content determined from image analysis of samples before and after melt extraction.

Figure 7.

Compaction curves for an olivine + MORB sample subjected to a pressure difference of 50 MPa at a temperature of 1476 K (C-450). The two volumetric strain records, calculated using (1) and (2), show good agreement in both absolute strain and strain rate. A small constant axial stress of about 1 MPa was imposed on the sample to obtain a continuous length record while avoiding notable axial creep. Oscillations in the length change record, such as the one starting after about 2000 s, occurred when the axial actuator “got lost” in the friction band.

[24] Measuring the melt content of samples as a function of distance from the interface between the sample and the carbon reservoir (Figure 8) suggests that melt viscosity affects the compaction process. Within the error of an individual measurement, the melt contents in the olivine + Li-silicate and olivine + MORB samples are spatially homogeneous. For clarity, only the samples with the lowest final melt fractions are shown in Figure 8. By contrast, there is a significant gradient in melt fraction in the olivine + albite glass sample. Indeed, the melt content at the undrained end of the sample remained unchanged.

Figure 8.

Melt fraction versus distance from reservoir/sample interface. Within error of measurement, there is no melt fraction gradient observed for the olivine + MORB and olivine + Li-silicate samples. In contrast, a significant gradient is observed for the olivine + albite glass sample. There was no change in melt content relative to the starting sample at the high melt fraction end of this sample.

[25] Since the compaction process is isotropic and homogeneous for all but the olivine + albite glass sample, the volumometer and the axial actuator records provide redundant information. The volumometer record is sensitive to temperature variations of the entire apparatus. Furthermore, leakage of gas from the volumometer to ambient pressure leads to an underestimation of volumetric strain, while leakage from the vessel into the pore pressure system leads to an overestimation of volumetric strain. Therefore, we used the axial displacement data to calculate the current melt content using (3).

3.3. Compaction Rates

[26] Compaction rates increase linearly with pressure difference and decrease with decreasing melt content. The volumetric strain rates reported in Table 3 were determined by linear regression of compaction curves over melt content intervals of 0.5%. The effects of melt content and pressure difference are illustrated in Figure 9 for an olivine + MORB sample with an initial melt fraction of ∼0.35. As shown in Figure 10, a compilation of data from all samples demonstrates that compaction rates change approximately linearly with pressure difference at a constant melt fraction.

Figure 9.

Results of a compaction experiment on an olivine + MORB sample (C-372) subjected to pressure differences of 6, 25, and 54 MPa. After an initial transient, compaction rates decrease with decreasing melt content at a given pressure difference and increase with increasing pressure difference at a given melt content. In this particular experiment, the pressure difference of 6 MPa was maintained while the sample was heated above the solidus. The initial acceleration of compaction rate is an artifact resulting from thermal equilibration of the sample assembly. Heating was performed while a nonzero pressure difference was applied and compaction started immediately after the temperature exceeded the solidus. In subsequent experiments we changed the procedure to avoid this problem. Heating was then performed at zero pressure difference. A pressure difference was applied about 30 min after the run temperature had been reached. This procedure ensures that the initial melt content is well constrained.

Figure 10.

The influence of pressure difference on compaction rate. The change in compaction rate observed after changes in the imposed pressure difference is well fit by a linear relationship; i.e., n = 1, where equation image.

Table 3. Compaction Rates as a Function of Melt Content
Exp. #FluxT (K)ΔP(MPa)ϕaequation image (s−1)
  • a

    Current melt content determined from the changes in length assuming isotropic compaction and using the melt content determined from image analysis.


[27] The effect of melt content on strain rate can be analyzed by normalizing the data to a constant pressure difference using equation imageV ∝ ΔPn with n = 1. The normalized strain rates from the three olivine + MORB experiments show good agreement (Figure 11a). Apart from transients at the beginning of the experiments, the data are well fit by a single power law relationship between melt content and normalized strain rate with an exponent of ∼3. At face value the experiments on the olivine + Li-silicate samples yield similar normalized compaction rates (Figure 11b).

Figure 11.

Compaction rates normalized to a pressure difference of 10 MPa using a linear relation between compaction rate and pressure difference. (a) Olivine + MORB samples; (b) olivine + Li-silicate (solid symbols) samples compared with olivine + MORB samples (crosses).

4. Discussion

[28] Two processes have to occur for a partially molten aggregate to compact permanently [e.g., Ahern and Turcotte, 1979]. The pore space must be reduced and the melt has to migrate out of the aggregate [McKenzie, 1984; Sleep, 1974]. In continuum models of melt segregation, the properties of the melt, the matrix, and the aggregate are combined into a scaling parameter called the compaction length [McKenzie, 1984; Ricard et al., 2001; Scott and Stevenson, 1986]. The compaction length is a characteristic scale over which gradients in fluid content, and therefore compaction rate, develop. Different estimates of the compaction length (δ) result from different model assumptions in the continuum approaches. The formulation of McKenzie [1984],

equation image

incorporates a bulk (ζ) and a shear viscosity (η) of the aggregate, the melt viscosity (μ), and the influence of melt fraction on permeability (k(ϕ)).

[29] In the Earth, compaction is primarily driven by gravity. By contrast, our samples are so small that the contribution of buoyancy to pressure gradients is negligible. To drive compaction, we imposed pressure gradients using the pore pressure system (Figure 1). Our experimental data determine the influence of melt on the viscosity and permeability of partially molten rocks, i.e., the parameters in (4). To calculate these properties we first need to analyze the relationship between imposed boundary conditions and the state of pressure in the samples.

[30] A fundamental observation for analyzing the pressure distribution in samples with low viscosity melts is that no significant gradient in melt content develops along the sample axis (x-direction) (Figure 8). Because there is no gradient in melt content in samples with final melt fractions ranging from 0.03 to 0.16, the condition ∂ϕ/∂x ≃ 0 is maintained through time. Applying conservation of mass, these observations indicate that melt velocity is constant along the sample axis and conservation of momentum for the fluid (Darcy's law) requires ∂2Pm/∂x2 = 0. We explore two end-member scenarios that satisfy these conditions.

4.1. Analysis of the Limiting Cases for Compaction

[31] One end-member is the trivial solution of a constant melt pressure identical to the reservoir fluid pressure, and corresponds to a perfectly drained experiment in which the resistance to fluid flow is negligible and compaction is limited by the rheology of the aggregate. A generalized law for the compaction rate of a porous matrix equation image can be written as

equation image

where A is a constant, f(ϕ,θ) is a function of the melt content and the dihedral angle which describes the local enhancement of the pressure difference and the changes in the rate-limiting diffusion length, Q is the apparent activation energy, and m and n are the grain size and the pressure difference exponents, respectively [Coble, 1970; Helle et al., 1985]. For comparison, the bulk viscosity used in two-phase theories is simply. equation image. Experimentally, ζ can be derived from hot-isostatic pressing experiments, as noted in appendix A of McKenzie [1984]. Such experiments demonstrate n = 1, indicating Newtonian behavior, when the pressure difference is relatively low [Coble, 1970; Helle et al., 1985].

[32] The second end-member is a constant pressure gradient. In this case, ΔP/l provides an upper estimate for the actual pressure gradient, as long as local dilation can be excluded. For fluid flux following Darcy's law, the volumetric strain rate accommodated by fluid transport equation image is

equation image

The relative importance of fluid transport and matrix compaction can be evaluated from an analysis of their characteristic timescales. The characteristic time required for a change in reservoir pressure to propagate a characteristic distance lc in an elastic porous medium can be estimated as equation image where βs denotes the storage capacity of the sample [Fischer and Paterson, 1989]. If the aggregate can compact viscously, the time required to relax the elastic strain owing to a change in pressure difference, εV,ela = βsΔP, is given by tc,mc ≃ βsζ. Accordingly (6) implies that the two characteristic times are similar, i.e., tc,mctc,ft.

[33] Since samples containing Li- or MORB-melt evolve through a succession of states characterized by homogeneous melt distribution, we consider the quasi-static relation between permeability and microstructure expressed by melt content and matrix grain size

equation image

The constant C quantifies the tortuosity of the paths on which fluid migrates. The melt fraction and grain size exponents (r and s) in (7) depend on the microstructure of the aggregate and its evolution [Bernabe et al., 2003; Gueguen and Palciauskas, 1994]. For olivine rocks where the melt topology is controlled by interfacial energy, numerical modeling yields r = s = 2 [van Bargen and Waff, 1986]. By contrast, experiments on texturally equilibrated quartz-water aggregates are best described by r = 3 and s = 2 [Wark and Watson, 1998]. In (7), the porosity should be specified as the connected porosity [Bernabé et al., 1982] or the difference between absolute porosity and a percolation threshold that depends on the dihedral angle [van Bargen and Waff, 1986].

4.2. Experimental Constraints on Melt Transport

[34] Analysis of the data for the olivine + MORB samples suggests that compaction is controlled by melt transport out of the sample as estimated by (6). First at all, recall that compaction rates increase linearly with pressure difference (Figure 10). In Figure 12a, we show the compaction rates for all three MORB experiments normalized to a constant grain size using s = 2 (c.f. equation (7)) and the final grain size of the samples. After the initial transients, these data fall within a factor of 2 to 3 of the strain rate predicted by (6) calculated using the MORB melt viscosity (Table 2) and the permeability measured by Wark and Watson [1998] for texturally equilibrated quartzites. For these calculations we used a tortuosity parameter C = 300, which deviates slightly from that reported by Wark and Watson [1998], but is appropriate when grain size is derived from intercept length multiplied by a geometric factor of 1.5 (D. A. Wark, personal communication, 2002). Our data also agree well with the predictions of (6) using permeabilities measured for Fontainebleau sandstone [Bourbie and Zinszner, 1985] after normalizing the grain size using s = 2 (Figure 12a). Grain growth during the individual compaction experiments, as quantified by the difference between starting and final grain size (Table 1), could lead to an underestimation of the exponent r by at most 25%. The fact that all three compaction experiments yield r ≃ 3 indicates a subordinate role of grain growth. Thus, our observations are consistent with permeability of ol + MORB aggregates determined by (7) with s = 2, r = 3, and C = 300 for a melt content range of 5 to 30%.

Figure 12.

Normalized compaction rates for samples with (a) MORB and (b) Li-silicate glass compared with compaction rates calculated using the relationship for transport limited compaction (equation (4)). Compaction rates were normalized to a pressure difference of 10 MPa, a sample length of 10 mm, and a grain size of 20 and 30 μm, respectively. Two permeability models were used to model the olivine data: k = (ϕ3d2)/C with C = 300 [Wark and Watson, 1998]; k = (ϕ2d2)/C with C = 1250 (van Bargen and Waff [1986]; the C-value was determined from their Figure 15 at a melt content of 5%). Additionally we show the permeability porosity relation for Fontainebleau sandstone determined by Bourbie and Zinszner [1985].

[35] In the olivine + albite glass sample, the olivine grain size is smaller than that in the olivine + MORB samples and the melt viscosity is much higher. Therefore, it is likely that the compaction of the olivine + albite glass aggregate was limited by melt transport, i.e., tc,mctc,ft. After a change in fluid pressure in the reservoir, melt pressure differences are only induced a short distance into the sample. Far from the sample-reservoir boundary, the melt pressure remains at the externally applied confining pressure. Thus permanent compaction is limited primarily to the vicinity of the sample-reservoir boundary. Ultimately, compaction may cease when the porosity at the boundary is so low that fluid flow is effectively inhibited (c.f., Figure 3 of Ricard et al. [2001]).

4.3. Experimental Constraints on Matrix Compaction

[36] The compaction rate of olivine + Li-silicate samples is more than an order of magnitude lower than that predicted using the melt-transport limit calculated from (6) (Figure 12b). In addition, the slope of the compaction rate versus melt fraction curve is significantly smaller than that observed for the olivine + MORB samples. At a given melt content, melt extraction in the olivine + Li-silicate samples is enhanced relative to that in the olivine + MORB samples by a lower melt viscosity and larger grain size. Thus, the disagreement between the prediction of the fluid transport limit and the observations suggests that matrix compaction is the limiting process.

[37] The conclusion that the strain rate of the Li-silicate samples is limited by matrix compaction is also supported by considering the influence of grain size on rheology. The compaction rates increase linearly with the pressure difference, i.e., n = 1 in (5) (Figure 10), suggesting that matrix compaction is controlled by melt-enhanced diffusive creep. At similar stresses and grain sizes, triaxial deformation experiments [Hirth and Kohlstedt, 1995b; Kohlstedt et al., 1999] and uniaxial compaction experiments [Cooper and Kohlstedt, 1984a] exhibit diffusion creep with m = 3 (i.e., Coble creep). Therefore, assuming that the deformation mechanism is the same for hydrostatic compaction, the matrix compaction rate for the coarser-grained olivine + Li-silicate samples is inhibited relative to that for the olivine + MORB samples. While the grain size difference is not large, normalization using m = 3 improves the agreement between the compaction rates determined from the two olivine + Li-silicate samples (Figure 13).

Figure 13.

Normalized compaction rates determined for olivine + Li-silicate samples compared to the initial stage hot-pressing model (equation (8)) and the modified Coble creep (equation (9) and appendix). We also show an exponential fit (equation imageV ∝ exp(αϕ)) to the compaction rates at melt contents below about 12%.

4.4. Strain Rate Enhancement Function

[38] The compaction rate of the olivine + Li-silicate samples can be used to constrain the form of the local enhancement function (f(ϕ,θ)) and therefore, the influence of melt content on the bulk viscosity of olivine aggregates. The relation between porosity and compaction rate during sintering and hot-pressing experiments has been extensively analyzed in the material sciences [Ashby, 1987; Coble, 1970; Helle et al., 1985; MacKenzie and Shuttleworth, 1949; McClelland, 1961; Murray et al., 1958]. This class of models accounts for changes in the geometry of grain contacts with decreasing porosity. At high porosities, the contact geometry is modeled assuming the interaction of spherical grains. Below a porosity of ∼0.1, the aggregate is represented by a continuum with spherical pores. By accounting for the change in coordination with decreasing porosity during the initial stage of compaction, Arzt et al. [1983] arrived at the following relationship between porosity and compaction rate limited by grain boundary diffusion (i.e., n = 1 in (5))

equation image

where ϕ0 is the initial porosity, Z0 is the initial coordination number, C is a geometrical constant and χ = (1 − ϕ0)/(1 − ϕ0). For an initial porosity similar to that for dense random packing of spheres (ϕ0 ≃ 0.36), Z0 ≃ 7.3 and C ≃ 15.5. As illustrated in Figure 13, (8) provides a good description for the rapid decrease in creep rate at the initial stages of compaction. In contrast, the influence of melt on compaction rate is significantly underestimated at melt fractions less than ≈0.1. However, considering complexities that are not included in these models (e.g., granular rearrangements, transient intracrystalline plasticity that occurs when contacts are established, variations in grain size), the agreement between the model and data is relatively good.

[39] Since these models were formulated to describe the compaction rate of single-phase aggregates, they do not specifically account for the rheological effects of melt with a small to moderate dihedral angle [Cooper and Kohlstedt, 1984b; Raj and Chyung, 1981]. In the appendix we derive a compaction law, in analogy with earlier work by Cooper and Kohlstedt [1984b] that described uniaxial deformation. This compaction law accounts for both the stress enhancement at grain contacts and the decrease in the rate-limiting diffusion distance that arise due to the presence of melt. The resulting constitutive equation is

equation image

where h(θ) is a function of the dihedral angle (θ) given in the appendix. The (1 − ϕ)−1 term accounts for increased local stress enhancement at reduced coordination. The second term in square brackets is a final stage compaction model applicable to moderate melt contents where the solid grains have constant and maximum coordination.

4.5. Comparison Between Experiments and Compaction Models

[40] As shown in Figure 13, the effect of melt fraction on compaction rate predicted by (9) at low melt fractions is only modestly greater than that predicted by (8), and significantly less than that observed in the experiments. The compaction rates predicted by (9) are rather insensitive to variations in the dihedral angle between 10 and 50° [Cooper and Kohlstedt, 1986].

[41] At face value, the discrepancy between the models and the data at melt fractions less than 0.1 indicates that the underlying geometrical assumptions used to derive (9) underestimate the effect of melt fraction on creep rate, i.e., an increase in melt content causes a larger increase in creep rate than predicted by (9). A similar conclusion is suggested by the results of triaxial deformation experiments [Hirth and Kohlstedt, 1995a, 1995b]. The effect of melt on creep rate could be enhanced by the presence of wetted two-grain boundaries that result from anisotropic interfacial energies [Faul et al., 1994; Hirth and Kohlstedt, 1995a]. In addition, variations in grain size produce local differences in coordination, even at small melt contents. Grain growth could also be responsible for an overestimation of the decrease of compaction rate with decreasing melt content. Furthermore, an overestimation of compaction rates at small melt contents would be expected if some of the porosity becomes disconnected, or if surface tension forces become significant. Neither of these latter possibilities seems likely given our knowledge of melt microstructure.

4.6. Constraints on the Compaction Length

[42] Our experimental constraints on the influence of melt fraction on the rheology and permeability of partially molten rocks can be used to calculate the compaction length. The form of the compaction length introduced by McKenzie [1984] (i.e., equation (4)) assumes that the bulk (ζ) and shear (η) viscosities constitute aggregate parameters, which depend on melt fraction [e.g., Scott and Stevenson, 1986]. By contrast, Bercovici et al. [2001] assume that the shear viscosity is an intrinsic and constant parameter of the matrix material. For this case, Ricard et al. [2001] arrive at

equation image

if k = k0ϕ3 and μ ≪ η.

[43] Assuming the compaction rate of the olivine + Li silicate samples in our experiments is limited by matrix compaction, these experiments provide a direct measurement of the influence of melt fraction on ζ. For the calculations, we use (9) to quantify the relationship between bulk viscosity and porosity. We describe the experimental relation between shear viscosity and porosity using an empirical exponential fit, η = η0 exp(−αϕ), where η0 is the shear viscosity of the material with no melt present, and α is in the range of 30–40 [Kelemen et al., 1997; Kohlstedt et al., 1999]. For comparison, we show how such an exponential relationship fits our compaction rate data in Figure 13. The magnitude of ηo is calculated using triaxial deformation data for diffusion creep of nominally melt-free olivine aggregates [Hirth and Kohlstedt, 1995b], where η0 = ηref (d/dref)m with η ref ≈ 9 × 1012 Pa·s at dref = 10 μm, T = 1473 K, and m = 3. The shear viscosity is assumed to be constant when calculating δBRS. Finally, we use the positive correlation of our olivine + MORB data to permeability measurements on analogous systems [Bourbie and Zinszner, 1985; Wark and Watson, 1998] to constrain k (i.e., k = d2ϕ3/300).

[44] The change in compaction length as a function of melt fraction is shown in Figure 14, together with a box representing the dimension of our samples and the melt fractions tested. Based on the microstructural observations shown in Figure 8, the compaction lengths of the olivine + MORB and olivine + Li-silicate samples should exceed the sample length for all melt fractions tested. By contrast, the observation that melt is extracted from ∼3/4 of the length of the olivine + albite sample indicates a compaction length on the order of 0.1–1.0 times the sample length. For the two lower viscosity melt systems, both the McKenzie [1984] and the Bercovici et al. [2001] models predict compaction lengths that exceed the sample length or are very nearly the same. For the olivine + albite melt sample, the compaction length predicted by the McKenzie [1984] model is ∼2 orders of magnitude smaller than the sample length. At face value, this small compaction length (∼0.1 mm) is at odds with the observation that at least some melt was extracted at distances up to 7–8 mm in this sample.

Figure 14.

Compaction lengths as a function of melt fraction calculated from models by McKenzie [1984]M, see equation (4)) and Bercovici et al. [2001]BRS, see equation (10)). The shaded box shows the melt fraction and sample length of our experiments. The approximate grain size of the samples investigated is given in the legend. Further details on the values of parameters used for these calculations are given in the text.

4.7. Extrapolation to Upper Mantle Conditions

[45] To investigate the geologic implications of our results, we calculate the compaction length as a function of melt content for partially molten olivine aggregates at upper mantle conditions. In the upper mantle, the olivine grain size is expected to be on the order of 1 to 10 mm [Evans et al., 2001]; the melt content is believed to vary from less than 1% in the source region to 3–12% just below the crust at mid-oceanic ridges based on geochemical evidence and geophysical surveys [Dunn et al., 2000]. For these ranges in grain size and melt fraction, our results suggest compaction lengths on the order of 0.1 to 10 km, with the lower estimate calculated for the smallest grain size and melt fraction.

5. Conclusions

[46] We investigated the extraction of melt from partially molten olivine aggregates by measuring compaction rates at a temperature of 1473 K. The viscosities of the three melts, fluxed either by Li-silicate, MORB, or albite-glass, ranged between 1 and 104 Pas. Observations of microstructure, changes in macroscopic sample geometry, and records of the external displacement transducers all indicate that the compaction process is isotropic and homogeneous for samples with lower melt viscosity. For the higher viscosity olivine + albite-glass sample, the melt fraction increases with distance away from the reservoir. For the two lower viscosity melts, no gradients in melt fraction were observed, even when the melt fraction was as low as 0.03. Compaction rates decrease with decreasing melt content ϕ or increasing melt viscosity, but increase linearly with an increase in the difference between the confining pressure and the melt pressure.

[47] The experimental data constrain permeability and bulk viscosity of the aggregates for the end-member cases where compaction is controlled either by fluid transport or matrix viscosity. Although the rates depend linearly on the difference between confining and fluid pressure, compaction rates for the two cases respond differently to changes in microstructure, melt content, and grain size. In the fine-grained olivine + MORB samples, permeability appears to be proportional to ϕ3 between melt fractions of 0.05 to 0.35 and to the grain size squared. The inferred permeability compares well with that measured for Fontainebleau sandstone [Bourbie and Zinszner, 1985] and fine-grained quartz aggregates with similar fluid topology [Wark and Watson, 1998]. Melt extraction from the coarser-grained, lower melt viscosity Li-silicate samples appears to be limited by viscous compaction of the solid matrix, thus constraining the bulk viscosity. At melt contents above about 10% the observed dependence of compaction rates on changes in melt content agrees well with a model for the initial stage of hot-isostatic pressing [Arzt et al., 1983]. If the partially molten aggregates are modeled as a composite of truncated Kelvin grains at lower melt contents, the effect of increasing melt content on the compaction rate is underestimated, a result also found in triaxial deformation tests [Hirth and Kohlstedt, 1995a]. When these results are extrapolated to conditions prevailing in the Earth's mantle we calculate a compaction length of about 1 km.


[48] Here, we derive a compaction law in analogy to the work by Cooper and Kohlstedt [1984b] for uniaxial deformation of partially molten aggregates. We assume the aggregate is composed of equi-sized Kelvin truncated octahedrons deforming by diffusion creep limited by grain boundary diffusion. The displacement rate of a stressed solid-solid contact is determined by the flux of particles migrating through its perimeter pss normalized to its area ass. The flux is proportional to the gradient in chemical potential Δμ/Δx between source and sink. The strain rate perpendicular to the contact is obtained by dividing the displacement rate by the distance (L) between contacts.

equation image

[49] The chemical potential of a contact is proportional to the confining pressure plus an additional stress (PcPm)amp/a that accounts for the stress enhancement resulting from the decrease in load bearing area owing to presence of melt, where amp = aass and a is the area of contact when no melt is present. The chemical potential of a melt pocket is proportional to the melt pressure Pm. An analysis of the creep rates of the two different types of faces on a “Kelvin grain”, octahedral (subscript of) and square (subscript sf) [Cooper and Kohlstedt, 1986], reveals that the displacement rate of the octahedral faces limits overall compaction. Assuming isotropic behavior the compaction rate is

equation image

where pof, Lof, and aof denote the perimeter, spacing, and area of octahedral faces of a Kelvin grain (subscript K) with a total surface area ak = Zofaof + Zsfasf. A prime indicates that a parameter describes the melt-present case. This geometrical model yields a melt-content dependence for the compaction rate of

equation image

with equation image and equation image [compare Cooper et al., 1989]. Since a constant coordination number is assumed in deriving (9), the model can be considered a final stage compaction model applicable to moderate melt contents. Equation (11) can be further modified using Arzt's [1982] approximation for the evolution of coordination number with porosity Z ≃ Z0(1 − ϕ). In this case, an additional factor of (1 − ϕ)−1 results from the local stress enhancement. We emphasize that the (1 − ϕ)−1 term is only a zero order correction, since the function h(θ) will also vary with the coordination number.