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 than as 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 for soil 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 Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/144/2/299/4888828/gsjgs.144.2.0299.pdf by guest on 02 October 2021 300 D. P. McKENZIE 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 in balance 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 already explained 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. Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/144/2/299/4888828/gsjgs.144.2.0299.pdf by guest on 02 October 2021 COMPACTION OF IGNEOUSSEDIMENTARYAND ROCKS 301 edge where three grains meet, for threevalues of Q.