The Compaction of Igneous and Sedimentary Rocks

Journal of the Geological Sociefy, London, Vol. 144, 1987, pp. 299-307, 7 figs. Printed in Northern Ireland

The compaction of igneous and sedimentary rocks

D. P. McKENZLE The Department of Earth Sciences, Bullard Laboratories, Madingley Rise, Madingley Road, Cambridge CB3 OEZ, UK

Abstract: In many problems of interest to geology and soil mechanics a fluid moves through a solid matrix which is also being deformed. In most cases of interest the effective viscosity of the matrix is many orders of magnitude greater than that of the percolating fluid, and both form interconnected networks in three dimensions. The partial differential equations governing such behaviournow can be obtained and a variety of simple model problems have been investigated. Some of the more surprising solutions contain solitary waves and compaction fronts. Some ofthe less exotic solutions are presently of more geological interest. The mobility of volatile rich melts at melt fractions as small as 0.1% has important consequences for trace element geochemistry. The geometry of layered intrusives appears to result from differential compaction of the crystal mush, and the rates required can be used to estimate a viscosity of 3 X 10" Pa S for an olivine matrix. This value agrees excellently with laboratory experiments when proper account is taken of the grain size dependence. Compaction in sedimentary rocks results in overpressure when the porosity becomes sufficiently small, and can also lead to the development of secondary porosity.

It has long been recognized that there are many geological exist between hydrocarbons and water. The presenceof two situations in which a fluid passes through and interacts with or more phases in the pore space fundamentally affects the a deforminga solid matrix. One such situation is the behaviour of the system. Despite its importance to the oil separation of magma from unmelted phases (restite) during industry, no satisfactory understanding of this problem yet partialmelting, another is the expulsion of theaqueous exists. Thecentral difficultyis thegeometry and phaseduring sedimentarycompaction. The same type of interconnectedness of thephases in three dimensions.As behaviouralso occurs in pyroclasticdeposits during the yet the physical principles which govern this geometry are expulsion of thegaseous phase. Most metamorphic not understood, although various attempts have been made geologists now believe that fluid movement is an important to guess the answer. The discussion below is therefore only process in regional metamorphism,and therefore this concerned with the problem in which one fluid fills the pore problem also requires an understanding of the relevant kind space. of two-phase flow. Anotherextremely important area in Thesecond problem concerns the nature of the which the same behaviour occurs is in soil mechanics, where framework through which the low viscosity fluid moves. In the mechanical properties of the soilmatrix are strongly many, though not all, problems of geological interest this affected by thepresence of water.Recently considerable framework is crystalline, and is therefore a deforming solid. progresshas beenmade in understandingthe fluid At high temperatures and low stresses the deformation of mechanics underlyingsuch phenomena, and the principal mostsolids can be described by a viscosity. Recently, purpose of this paper is to review this work. Much of what Cooper & Kohlstedt (1984) havedemonstrated that the hasbeen published is of considerablemathematical olivine matrix deforms in this way when the pores are filled complexity,and in thisform is not easily applied by with basaltic melt, and similar behaviour is to be expectedin geologists. But many of thecentral ideas and results are other situations when the shear stress is small. When the rather simple anduncontroversial, and allow important stress is larger thedependence of creeprate onstress is conclusionsto be drawnabout problems of geological unlikely to be linear, and the appropriate equations will be interest.It is theseareas with which this review will be thosefor thedeformation of non-newtoniana fluid. concerned. No attempt is made to derive the results which Nonetheless,the matrix will generallybehave as a fluid are used. Such derivations will be found in the publications rather thanas a solid. The expulsion of the lowviscosity referenced. fluid fromthe pores is therefore generallyan irreversible The first problem in discussing two-phase flow is one of process, and the resulting stresses in the matrix are relaxed terminology. The low viscosity phase is variously described by creep processes. Thedeformation of the matrix must as a gas, a melt, an aqueous phase, a fluid or a vapour. By therefore also be described by the equations of fluid flow. implication these terms are used to describe different types In soil mechanics a different assumption is usually made, of material filling thepore space. But thisdistinction is andthe matrix is treated as anelastic solid. The artificial, andthe physical behaviour of the system is mathematicalrelationship between the two approaches is controlled by the fluid properties.The terms melt and briefly discussed below. One of the difficulties of soil interstitial fluid will be used interchangeably to describe the mechanics is that the assumption of elasticity leads to an low viscosity material filling the pore space. In contrast to underestimate of the time required forsoil consolidation, this issue, which is one of terminology only, is the question and a more realistic rheology should probablybe used. of whether there is more than one low viscosity fluid phase Manyproblems therefore require an understanding of present.Two, sometimes three, phases are present in oil the behaviour of a material which consists of interconnected and gas reservoirs, as a result of the immiscibility gaps that networks of each of two phases. The deformation of both 299

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can be described by a viscosity, but we will assume that the of the melt and matrix for more than 30 years (Smith 1948; viscosities of the two phases are very different (they typically Kingery et al. 1976). The central concept is that of textural differ by a factor of morethan 10'"). The high viscosity equilibrium; this is attained when the surface tension forces material will bereferred to as the matrix, that of low are inbalance wherever fluid and crystals meet. Figure 1 viscosity asthe melt or interstitial fluid. For the reasons shows a typical section of a junction between two crystals alreadyexplained no attempt will be made to discuss the and melt where ymm is the surfacetension between two behaviour of systems containing two interstitial phases. crystal grains and ymf that between a crystal andthe The discussion below consists of three parts. The first, interstitial fluid. The section shown is normal to the grain phase geometry, is concerned with the geometry of the melt edge. Resolving forces along the boundary between the two and matrix. The second, deformation of melt and matrix, grains shows that outlinesthe ideas behind the derivation of the fluid dynamical equations and the interpretation of the constants and variables they contain. A variety of solutions to simple model problemsare also discussed here.The third part, and normal to the boundary requires geological problems, is concerned with some geological applications of these ideas. LY1 = m* 0 is known as the dihedral angle, and depends only on Phase geometry the surface tension and hence thesurface energies; although Material scientists have had a general understanding of the most authors define 0 using equation (l), a few, such as processes and general principles which control the geometry Beer6 (1975), insteadrefer to LY as the dihedral angle. Equation (1) shows that the geometry of all grain edges is controlled by a single scalar quantity Q. If Q is given it is clearly possible in principle to calculate the geometry of the melt phase for any melt fraction when the matrix grains all have the same shape.The size of the matrix grains then does not affect their shape, since the geometry can be magnified Matrix by any factorwithout changing Q orthe melt fraction. Although this calculation is in principle straightforward, it is complicatedin practice by the difficulty of describing the shape of the interfaces in three dimensions. Two solutions

A for different geometrieshave been published by Beer6 (1975) and by Park & Yoon (1985) for differently shaped 8 mm matrix grains. Park & Yoon argue that BeerC's calculations arein error, and that the total interfacial energy of the system increases more rapidly with melt fraction than his Matrix curves suggest. Two-dimensional analytical results (D. McKenzie in preparation) support Park& Yoon's argument. The details of the three dimensional melt geometry are of interest,and need to be investigated systematically; Fig. 1. The dihedral angle 0 is related to the surface tensions at the however, by farthe most geologically important result matrix-matrix ymmand matrix fluid ymfby concerns the condition which must be satisfied if the melt network is to be everywhere interconnected. This condition cos-=-@ Ymm is that 0 <60°, as was first pointed out in the geological 2 2Ymf literature by Waff & Bulau (1979). It is easy to understand and at = cr2 how this result occurs; Fig. 2 shows a section through the

(C)

M M / M / /

@ < 60' 8 = 60° 8 > 60' Fig. 2. (a) When 0 < 60" the fluid pressureis less than that in the matrix and the pressure difference increases as the fluid cavity becomes smaller.(b) When 0 = 60" there is no pressure difference between fluid and matrix. (c) When 8 >W the fluid pressureexceeds the matrix pressure, and the difference increases as the cavity becomes smaller.

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edge where three grains meet, for threevalues of Q. Clearly fluids in equilibrium with metamorphic and igneous rocks of there is a pressure difference AP between the matrix and the crustal compositions. fluid, where The rest of the discussion below is concerned with the 2 Ymf movement of the melt and matrix when < 60", and does AP=- (2) 0 R not take account of any influence the surface energies may have on this behaviour. In the laboratory such effects can be and R is the radius of curvature of the interface. In Fig. 2(a) important because grain sizes of a few microns are the fluidis ata lower pressure thanthe matrix, and this commonly used, and the size of the experimental apparatus difference increasesas the size of the melt channel,and is generally small. At such grain sizes surface energy effects hence R, decreases. In equilibrium, however, the pressure can dominate gravitation effects when the vertical scale is in the fluid will be the same everywhere, and smaller melt less than a few metres. In geological situations the scales are channels will grow and larger ones shrink until they all have larger and grain sizes are 1 mm or more. Under these the same curvature. Clearly it is not possible in this case for conditions the surface forces are only important for layers = anywhere until all the melthas been removed R 0 whose thickness is less thana few centimetres.Their entirely. Thereforethe meltnetwork must remain influence onthe dynamics can therefore be neglected in interconnected throughoutthe material, no matter how most situations (D. McKenzie in preparation). small the meltfraction may be. Geologists often describe If melt and matrix are to reach textural equilibrium there this interconnectedness as being due to wetting of the grains must be some process by which the grains of the matrix can by the melt. This is not correct. Wetting occurs when change shape.The most important process is probably Ymm > 2 Ymf (3) transport by diffusion in the melt phase. The internal deformation of the grainsthemselves, driven by surface There is no solution to equation (1) for 0,and all the grains forces,must also occur,but is likely to be considerably are covered by melt andcannot touch.Clearly, since slower than diffusion through the melt.Although no 0 < 60", the melt network is interconnected. But 0 = 0 is detailed measurements have yet been carried out on the rate not a necessary condition for interconnection, and, as Arndt at which textural equilibrium is established, estimates can be (1977a,b) and Walker et al. (1978) have remarked, it is madefrom the results of laboratoryexperiments and harder to extract small melt fractions from the films on grain geological observations onthe basalt-olivine system. The boundaries which exist when 0 = 0 than from thetube experiments of Waff & Bulau (1979), Vaughan et al. (1982) system which is stable when 0 < 0 < 60". andCooper & Kohlstedt (1984) show that textural When 0 = 60" the matrix-melt interfaces are flat and equilibration occurs for grains whose size is between 5 and R = m. Then AP = 0 and there is no tendency for the melt 10 p inatime of less than 200 h. If the process is channels to grow or shrink. diffusion-controlled it should scale as the square of the grain When 0 > 60" the pressure in the channels is greater radius. A typical geological grain size of 1 mm then than that in the matrix, and increases as R -+ 0. Hence small corresponds to atime scale of about 103 years, whichis channels collapse and isolated poresare formed. This small compared to the cooling time of largeintrusions. condition requires Hence texturalequilibration would beexpected provided 1 the leastdimension of such bodies exceeds about 1 km. Ymf > 3Ymm (4) This estimate agrees with Hunter's observations (Hunter 1987) that textural equilibration commonly occurs in large and most commonly occurswhen thepore space is filled igneous intrusions and during regional metamorphism. with agas, such asair. A problem of major interestin The process by which the equilibration occurs is by materials science is the densification of high melting point dissolution where the surface energy is high and deposition powders at high temperatures. Thisprocess is known as where it is low. If, however, the interstitial fluid is sintering, and is driven by surface forces. It can only occur supersaturated in the matrix component, only deposition when 0 > 60", which is unfortunately also the condition will occur and textural equilibration will not take place. The required for the formation of isolated pores. Much research petrography of most compactedsediments shows that has therefore been directed to finding the conditions under texturalequilibration frequently doesnot occur in such which maximum densification can occur. environments.Kaolinite andother clay minerals are As this discussion illustrates, the question of whether the exsolved with shapes which havelarge surface areas, melt network is interconnected depends on the value of 0 presumablybecause the fluids responsible are strongly which therefore governs whether small melt fractions can be supersaturated. Calcitecements also often fill pre-existing extracted.Unfortunately onlya few such measurements pore space and must therefore have been transported from have yet been carried out on different melt compositions, elsewhere. Only where there has been extensive pressure despite their importance. Waff & Bulau (1979) showed that solution is texturalequilibration likely to have occurred. basaltic melt in contact with olivine had a dihedral angle of Hencethe complicatedgeometry of interstitialpores in about 50". Jurewicz & Watson (1984) found that 0 = 60" for sedimentaryrocks is likely to be governed more by the aquartz-albite melt in a quartz matrix. R. H. Hunter & dynamics of crystal growth than by surface energy. D. McKenzie (in preparation) show that a melt principally In contrast to those of sedimentary rocks, many of the composed of dolomite in an olivine matrix has a value of 0 textures of metamorphic rocks and of largeigneous which is less than 60", whereas T. Andersen (pers. comm.) intrusions show clearindications of texturalequilibration has found that the dihedral angle for CO, inclusions in a (Hunter 1987). The angles between grains of different clinopyroxene matrix is about 100". Extensive measurements minerals correspond to those expected for constant surface of 0 are required, especially for some of the more exotic energies, andthe geometry of the phasesfrequently melt compositions which are sometimes erupted,and for attemptsto minimize the surfaceenergy. Whether such

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texturesare producedwhen the melt is still present or strongly indeed. If in addition the movement of the melt is whether they result from extensive subsolidus flow has not required to satisfy D'Arcy's law only one formfor the yet been established. For our present concern this question interaction term is possible and the governing equations are is not of primary importance. If the textures are produced obtained.Various other approaches to this problem have when the fluid phase is present, they indicate that textural been attempted, (Scott & Stevenson 1986; A. Fowler equilibration occurs under such conditions. If they are the unpublished manuscript) and these other equations can be result of subsolidusprocesses, they indicate that textural obtained as approximations to those obtained by McKenzie. equilibrationoccurs sufficiently rapidly to affect the grain However, none of theother formulations are strictly shapes evenwhen nomelt is present,and hence will invariant to gallilean transformations,although the certainlyoccur at higher temperatures in the presence of differences from McKenzie's equations are small under melt. The observationsshow, therefore,that textural most circumstances, and appear not to affect the solutions. equilibration occurs when the cooling rate is slow, in large It is important to understand the assumptions made in intrusions or in the regions from which melt is extracted. using mixturetheory. The smallest region of interest is assumed to be large compared with the size of individual grains, yet small compared with the distances over which the The deformation of melt and matrix velocities changes. Boththe melt andthe matrix are considered to be fluids which fill all space but interact with each other. Their velocities are the averages of those of the The governing equations matrix and melt, but cannotdescribe the detailed flow of the The most generaland compactdescription of a physical melt or the matrix onthe scale of individual grains. process is provided by the governing differential equations. Compaction of the matrix is taken into account by allowing Once these have been obtained, solutions can be found for the matrix to be compressible, with a bulk modulus of zero. different boundary conditions and their stability investigated Instantaneous collapse is avoided by including a bulk by discovering whether small disturbances grow or decay viscosity, which producesa viscous resistance to isotropic with time. Such studies canbe carried out numerically, collapse. The resulting equations satisfactorily describe the analytically or by using properly scaled laboratory velocities of the melt and matrix and the pressure within the experiments. melt. There is, however, no simple correspondence between The first attempt to approach the compactionproblem the isotropic part of the stress tensor within the matrix from this point of view was made by Sleep (1974), but little obtained from mixture theory and the average microscopic use was made of the equations he obtained. Earlierwork on pressure within the matrix. what was known to igneous petrologists as filter pressing was The use of mixturetheory also allows the standard largely intuitive and is summarized by Propach (1976). One equations of soil mechanics to be derived(Biot 1941; of theseearlier ideas which is still widely believed is that Terzaghi 1925 (seeTerzaghi & Peck 1967)) ina manner shearing of the matrix in some manner aids compaction. In which is perhaps simpler than that used by Biot. If, instead fact neither simple shear nor pure shearof the matrix affects of requiring the matrix to deform as a viscous fluid, it is the flow of the melt. To do so the matrix velocity must have required to satisfy the equations of an elastic compressible a divergence or a non-zero second derivative. solid, Biot's equations result (D. McKenzie, in preparation). Sleep's (1974) early work on this problem contains one More complicated matrix rheologies, such as those of Kelvin error, which leads to an infinite compaction rate under an or Maxwell solids, can also be described by the same isotropic stress field. Although this error is important, and approach, ascan the non-linearbehaviour typical of high would have invalidated attempts to apply his equations, the temperature, high stress, creep governed by dislocation basic form of his equations is otherwisecorrect and this movement. Mixture theory therefore provides a framework error is not difficult to correct. for obtaining self consistent groups of equations which can The general form of the governing equations which did describecomplicated solid rheology, which may be of not suffer fromthis problem was obtained by McKenzie considerable value in soil mechanics. (1984), using rather simple physical arguments. In common with Sleep he argued that the dynamics of the matrix and Solutions the meltshould be considered separately, as if each occupied all space. Each must obeythestandard Although theequations governing compaction have only conservation laws of mass, momentum, andenergy. The beenrecently discovered, several problems of geological interaction of each fluid with the other is taken into account interest have been investigated. using a transport term, with the condition that mass, vector The simplest problem is the behaviour of a layer with momentum or energy lost from one fluid must at once be constant melt fraction when placed onan impermeable gained by the other, and hence conserved. The basis of this surface. If the matrix is denserthan the melt, the matrix approach is explained in Ishii (1975), and is known as compacts and expels the melt to form a layer on top. The mixture theory.The transport terms must satisfy various initial behaviour of this system is somewhat surprising. If the conditions. Their value clearly must not dependon what layer is sufficiently thick,compaction initially only occurs frame of reference is used to describe the flow; if it did the within a boundary layer next to the impermeable surface. behaviour would dependon whether the observer was The thickness of this layer is controlled by the compaction stationary or moving with respect to the matrix, and such length 6, contradictionsmust be excluded.Although this condition, 6, = + a)S/pI'" (5) known theas invariance with respect to gallilean [(k : transformations, is simple, it is sufficient to restrict the form where and arethe bulk and shear viscosities of the of the interaction terms in themomentum equation very matrix, p the shear viscosity of the melt and k+ the

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(a) (b) 4- Compactionrate

Matrix velocity Melt W' velocity - W' .

I L -0.1 1 - 1 0

Q, = 0.1

Fig. 3. (a) The velocity of the melt W' and the matrix W' when a porous layer of porosity of 0.1 is placed on an impermeable surface atz' = 0. The velocity is taken to be positive in the +z' direction. The true velocities are wow' and wow',and the true height z = &z'. (b) The instantaneous compaction rate produced by the velocities in (a). a+/&' is negative because compaction is occurring, and the true time t = rot'

permeability of the melt network. The formof equation (5) is with the numerical results (McKenzie 198%) is surprising, because it is independent of the density contrast Ap between melt and matrix. Above the boundary layer the weight of the matrix is supported by the upward percolation of the fluid expelledfrom thecompacting region. It When h >> 6, substitution gives percolates upward with just sufficient velocity W, to support the matrix, where h t- hP+ h -wo(l - 4) =(l - +)'kSApg (9) WO = b(1- +)&%/P+ (6) g is the acceleration due to gravityand the melt fraction Hence, c), is the independent of the mechanical properties of by volume or porosity.This initial behaviour is illustrated in the matrix. This result was obtained earlier by Stopler et al. (1981), who argued that the force required to deform the Fig. 3, and provides a natural time scale tofor the problem; the time for the porosity at the base of the layer to change matrix in thick layers of partial melt was small compared by a factor of e at the initial compaction rate with thepressure gradient required to move the melt through the channels. A similar argument was also proposed by Ahern & Turcotte (1979). L(*) = -wo(l - +)/S, = --1 The other limiting case occurs when h << 6,, when 4) at *=o TO The simpleanalytic solutionobtained by McKenzie (1984) and illustrated in Fig. 3 is only valid at t = 0. As the system evolves thesupply of meltfrom the compacting and the behaviour is independent of the permeability and region becomes depleted and compaction spreads upwards. viscosity of the melt, being controlled by the flowof the The problem can then no longer be solved analytically, but matrix alone. This behaviour was also discussed by Stopler numerical solutions are easily obtained. Two questions are et al. (1981). There is thereforea satisfactory agreement of interest: how quickly is the melt expelled, and how does between the behaviour expected using physical arguments the porosity distribution evolve? A suitable measure of the and that obtained from numerical solutionsof the governing time-scale is the time C,, required for the melt fractionin the equations. layer thickness h to decrease by a factor of e; from h$ to The variation of the porosity with time in the problem h+/e. A simple empirical expression for th which agrees well discussed above is illustrated in Fig. 4a, taken from Richter

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sedimentology and to the igneous petrology of large magma chambers, and is being investigated insome detail (D. 50 McKenzie in preparation). As the material is buried, its rate of compaction is at first controlled by therate of deformation of the matrix. The permeability is sufficiently

40 large for the interstitial fluid to be expelled easily, and the pressure gradient is hydrostatic. However, at a critical depth the porosity becomes sufficiently small for the movement of the interstitial fluid to be important. The pressure gradient 30 increases in this region by a factor of 103 or more, and the 2' pressure increases from hydrostatic to the lithostatic over a small depth interval. Beneath this zone, which will be 20 referred to as the compaction front, a number of solitary waves are formed which propagate upwards,though at a velocity which is less than that of the compaction front. The

la movement of these waves involves small local increases in porosity, and in these regions the pressure must exceed the lithostatic pressure. Some of these features are illustrated in Fig. 5. From the geological point of view the most important 0 0. l parameters are the depth of the compaction front below the CD 9 upper surface, z,, and the total thickness of interstitial fluid Fig. 4. (a) The evolution of the porosity with time for a layer whose tf between thefront and the upper surface. Simple initial porosity is 0.1 everywhere, as in Fig. 3, but whose approximate expressions for these are dimensionless depth is 50 insteadof 4. The numbers against the curves show the times in unitsof z,. (b) As for (a) but with an initial porosity shown by the solidline. A train of solitary waves results after a time 18 z,. (b)

& McKenzie (1984), and is not surprising. Thedeeper l8 regions are always the mostcompacted, andthe porosity increasesmonotonically towards the surface.If, however, the initialporosity anywhere decreases with height the behaviour is quite different.Solitary waves are produced which propagate upwards at a velocity which dependson 16 their amplitude (Richter & McKenzie 1984; Scott & Stevenson 1984). Figure 4b illustrates this behaviour, which has generated considerable interest among applied mathe- maticians. It is possible to find a porosity distribution which 14 propagates with unchanging form, rather than breaking up into a series of solitary waves (Richter & McKenzie 1984; Z' Barcilon & Richter 1986), and so thereare true solitary wave solutions to this problem. These waves do not appear

to be solitons, because they do not pass through each other 12 with unchanging form (Barcilon & Richter 1986). Scott et a1 (1986) have carriedout some beautifulexperiments to illustrate the propagation and interaction of these solitary waves, and Scott & Stevenson (1986) have demonstrated that one-dimensional solitary waves in a compacting system 10 are unstable to two-dimensional disturbances. Whether they are alsounstable to three-dimensionaldisturbances, and what the planform of the stable flow is, are questions which 0 0.5 have not yet been answered. Q, P- P, These fluid dynamical resultshave important implica- Fig. 5. (a) The porosity distribution producedby a constant tions forthe geological problems.Wherever the per- sedimentation rate with a dimensionless sedimentation rate of0.2 meabilitydecreases upwards, waves of increased porosity onto the upper surface of a compacting layer, initially atz' = 10, will form and propagate upward breaking up into vertical and of porosity 0.05, after a timeof 37.5 z,. The dashed line shows sheets, like dykes, as they move further upward. There is an 10 #(z). (b) The difference between the fluid pressure and obvious correspondence with the geometry of igneous hydrostatic pressure, called the piezometric pressure, in arbitrary intrusions and with that of veins in sedimentary rocks. units, corresponding to the porosity distributionin (a). The dotted The compaction problems discussed above are not line shows the corresponding curve for the total static pressure concerned with the sedimentation problem, where material which the material above could exert, called the lithostatic pressure. with high andconstant porosity is addedto the upper Notice the piezometricpressure exceeds the lithostatic pressure surface of the layer and compactsas it is buried. This where the porosity is increasing at the front of the solitary waves, problem is of considerable geological interest toboth which are moving upward.

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and GARNET PERlDOTlTE l I I l I l l I

where W, is the sedimentation rate and the surface porosity has been assumed to be 30%. These expressions allow the matrix viscosity to be estimated directly from the geological observations.

Geological problems From the geological point of view probably the single most important result of the work describedin the last section concerns the melt fraction likely to be present during partial melting of mantle materials. All authors who have examined 10.~ \ this problem (Ahern & Turcotte 1979; Stopler et al. 1981; yy I I I l I I l I McKenzie 198%)have found that basaltic melt fractions 0.4 0.8 1.2 1.6 greaterthan about 2% rapidly separatefrom the olivine IONIC RADIUS A matrix. When the melt is volatile-rich separation can occur 6. Observed (0)and predicted bulk distribution coefficients even when the melt fraction is as small as 0.1%. Despite Fig. (I) between rock and meltfor a varietyof elements, plotted as a rather extensive geochemical evidence that melt moves with function of ionic radius (Onuma et al. 1968), for a varietyof respect to the matrix when the melt fraction is considerably elements of different valencies,shown in the circles as ionic charge. less than 1% (see for instance Thompson et al. 1984, Fitton & Dunlop 1985), there is neverthelessa commonly held belief amongigneous petrologists that movement is only possible when the melt fraction exceeds 5 or 10%. This view previously proposed, and velocities as large as 100 mm yr-' seems to be primarily intuitive, and I have not been able to are still consistent with the observations. However, because find references to where it originates. I amtherefore not of the atypical composition of the melt involved, this able to discuss the arguments used in its support. estimate of $I is not likely to apply to the region undergoing One particularlystraighforward argument in favour of extensive silicate melting, where the porosity is likely to be the movement of small meltfractions dependson the as large as l or 2%. uranium decay series (McKenzie 19856). The activity ratio This argument outlined above has been criticized by C. r = (23@Th/238U)should be unity if radioactive equilibrium J. Allkgre (pers. comm.) who argues that the melt is not has been established, since each radioactive species in the likely to be in chemical equilibrium with the matrix. Since decay seriesmust then decayequally often. Yet the time required for 1mm grains to equilibrate by solid measurements of this ratio from many fresh MORBS have state diffusion is unlikely to be greater than 103years, during found an averagevalue of about1.25 (Allegre & which time the matrix will only upwell 100 m at a velocity of Condomines 1982; Newman et al. 1983). A simple 100mm year-', it is not easy to understand how the melt explanation of thisobservation is thatthe melt fraction could beproduced sufficiently rapidly to avoid equilibra- present is sufficiently small to allow 230Th and 238U to tion. Furthermore if it did so, the activity ratio should fractionate differently between the melt and the matrix. The approach unity, since almost all the U and Th should be in melt fraction $I present at any time can then be calculated the melt. directly from r and the distribution coefficients KD and K, of The meltfraction present during extensive silicate Th and U melting at shallow depthsbeneath ridges is not easy to estimate from observed isotopic or elemental abundances in MORB.The observed major element compositions are, however, compatible with direct extraction of melt from a It is independent of the Uconcentration. No accurate garnet peridotite source region (D. McKenzie & M. Bickle experimental estimates of K, and KD are available, but the in preparation). observations do provide an upper bound of KPe 0.005 for a From the point of view of trace element geochemists the garnetperidotite (McKenzie 1985b). The suggestion of mobility of melt fractions as small as 0.1% is of great Onuma et al. (1968), thatthe distribution coefficient is importance. The movement of such melts will dominate the correlated with the ionic radius, can be used to estimate K behaviour of the very incompatible elements, and studies of for Th and U (Fig. 6). The values obtained are 3.4 X 10-3 their distribution should illuminate the processes involved. and 1.5 X 10-3 for U and Th respectively fora garnet It is not, however,relevant tothe problem of extensive peridotitematrix. Substitution intoequation (13) gives melting which produces large volumes of magma. A good $I = 0.6%. Melt fractions as small as this are only mobile if rule of thumb is that elementaldistributions are strongly they are rich in volatiles (McKenzie 1985a). It is therefore affected by degreea of melting equal to their bulk likely that most of the U and Th in the mantle are extracted distribution coefficient. HenceTh and U are affected by by the small fractions of volatile rich melt produced at the 0.1% melting. The investigation of major amounts of onset of melting. Such behaviour would also be expected on melting requires studies of elements with distribution chemical grounds.Because such melting occurs over a coefficients between0.1 and 1. Apart from themajor vertical distance which is probably as great as 100 km, the elements, such elements have not had theattention they constraint onthe upwelling velocity is weaker thanthat deserve.

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Whether the existence of solitary waves in two and three detailed correspondence between the sedimentary problem dimensions within meltingcompacting regions have direct and the simple one dimensional model outline above. observational consequences is not yet clear. The expected The compaction of the mushy zones atthe base of planform of the melt channels is not yet known, and any magma chambers has been less extensively studied, but it is surfaceexpression must be strongly modified by the cold a simpler problem than sedimentary compaction because the brittle layer at the earth's surface. grain size is less variable and the textures are equilibrated. In contrast, compaction in systems undergoing sedimen- Irvine (1980) suggested that such compaction was the tation has been widely studied in sedimentary environments, explanation of the striking synclinal form of layering in and the pressuredistribution is also known from drilling. many large igneous intrusions. His suggestion is illustrated The observed behaviour (Magara 1978) is in general similar in Fig. 7 and depends on theinterstitial melt near the side of tothat expected fromthe simple one-dimensionalmodel the intrusion freezing before it can be expelled, whereas in discussed in the last section. At shallow depths thepressure is thecentre the melt is removed by compaction. Inthe hydrostatic, but often ceases to be so at depths greater than Rhodesian Great Dyke thelayering involves chromite bands 3 km (Magara 1978, fig. 2-21). Pressuresbetween hydro- which have been extensively mined and whose geometry is static and lithostatic values are known as overpressures to therefore particularly well known (Worst 1960; Prendergast petroleum geologists, and are commonly observed in deep, pers. comm.). Furthermorethe synclinal shape of the thoughnot in shallow, wells. Magara 1978, fig.2-224 also layering is strikingly uniform along the length of the 'Dyke'. shows that the porosity at a given depth increases with the If Irvine's suggestion is correct, the thickness of melt sedimentation rate, a relationship also to be expected from expelled in the central region tf can be estimated from the the one-dimensionalmodel discussed above.But the difference in elevation of the chromite bands between the variations in grain size in a real sedimentary basin produce margins andthe axis to be about 0.5 km. Irvine (1980) enormous variations in permeability, which must also be estimates the sedimentation rate insuch magma chambers to anisotropicbecause of the sedimentarylaminations. It beabout 300 mm year-'. Equation (12) then allows the would thereforebe naive to believe thatthere will be a effective viscosity, 5 + !v, of matrix to be calculated to be about 3 X 10l8Pa S. This value should be compared with that obtainedfrom uniaxial compaction experiments in the laboratory carried out by Cooper & Kohlstedt (1984). They Melt used grain sizes between 3 and 13 pm in order to establish textural equilibrium, and found that the effective viscosity varied with the cube of the grain size. If their results are I chromite I scaled to a grain radius of 0.5 mm and a temperature of 1300 "C, the average viscosity of the matrix is 8 X 10" Pa S. There is therefore goodagreement between these two estimates.

Discussion The progress made in the last 5 years in understanding the physical processes involved in compaction, and formulating the governing differential equations, has already produced a number of results of interest to geologists. Some of these are both important and surprising. The mobility of volatile-rich melts when the meltfraction is as small as 0.1% is unexpected. Solitary waves and compaction fronts were only discovered when the governing differential equationshad been properly formulated, and neither has yet been directly observed. The equations probably possess other unexpected types of solutions in two and three dimensions. Another problem of considerable interest concerns the dynamical interaction of matrixdeformation with melt movement. The simplest model problems,in which the matrix is deformed by pure or simple shear, do not produce results of interest. But beneath ridges and trenches the flow cannot be described as pure or simple shear, and the second spatial derivative of the matrix velocity is not zero. There is Fig. 7. Sketch to illustrate the mechanism suggested by Irvine thereforean interactionbetween the matrix deformation (1980) for producing the synclinal structure in the Great Dyke and and melt migration, whose investigation may lead to results other layered intrusions by compaction. Layers of chromite and of geological interest. other minerals are initially deposited as horizontal layers at the The most obvious area whereexperiments are required surface of the partially molten zone. Subsidence occurs in the axial concerns the measurement of dihedral angles. It is already region where the melt is expelled by compaction, but not at the clear that 0 depends on the composition of the melt and margins where it solidifies before it has time to be expelled. matrix, and that changes in composition of the interstitial Expressions for t, the total thickness of the melt expelled, andz, the fluid can cause 0 to change from less than 60" to greater thickness of the partially molten zone, are given in the text. than 60°, with potentially major consequences. Extensive

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measurements of this important parameter are needed, for KINGERY, W. D., BOWEN, H. K. & UHLMANN,D. R. 1976. Introduction to both igneous and metamorphic systems, and are not difficult Ceramics. Wiley-Interscience, New York. to carry out.Experiments such as those of Cooper & MCKENZIE,D. 1984. The generation and compaction of partially molten rock. Journal of Petrology, 25, 713-65. Kohlstedt (1984) onthe dynamics of meltmovement are - 1985a. the extraction of magma from the crust and mantle. Earth and harder,but are of great valueinunderstanding the Planetary Science Letters, 74, 81-91. microscopic and macroscopic processes at work. - 19856.23%-u8U disequilibrium and the meltingprocesses beneath As this short review illustrates, a new andimportant ridge axes. Earth and Planetary Science Letters, 72, 149-57. MAGARA,K. 1978. Compaction and Fluid Migration. Elsevier, Amsterdam. branch of geological fluid mechanics has been discovered. NEWMAN,S., FINKEL,R. C. & MACDOUGALL,J. D. 1983.2%-238U Though the governing equationsare non-linear,they are disequilibrium systematics in oceanic tholeiites from 21"N on the East tractableto both analytic and numericaltechniques, and Pacific. Earth and Planetary Science Letters, 65, 17-33. have already yielded results of major geological interest. ONUMA,N., HIGUCHI,H., WAKITA,H. & NAGASAWA,1968. H. Trace element partition between two pyroxenes and the host lava. Earth and Planetary Science Letters, 5, 47-51. I would liketo thank V. Barcilon, S. Galer, R. Hunter, K. PARK, H. H. & YOON, D. N. 1985.Effect of dihedral angleon the O'Nions, M. Prendergast, F. Richter, D. Scott, D. Stevenson and morphology of grains in a matrix phase. Metallurgical Transactions, 16, 923-8. D. Walker fo their help. This workwas supported by grant PROPACH,G. 1976. Models of filter differentiation. Lirhos, 9, 203-9. GR3/5607 from N.E.R.C. Contribution 813 of the Department of RICHTER,F. M. & MC~NZIE,D. P.1984. 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Received 13 May 1986; revised typescript accepted 13 October 1986.

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