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Determination of the rate-limiting mechanism for quartz pressure dissolution
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Gewhimica EI Cosmorhimica Acla Vol. 57: pp. 1499-I 503 Copyright 0 1993 Press Ltd. Pnnlcd in U.S.A.
Perl~arnon
Determination of the rate-limiting mechanism for quartz pressure dissolution ANDREW MARTIN MULLS* Department of Geology and Applied Geology, University of Glasgow, Lilybank Gardens, Glasgow G 12-SQQ, Scotland (Received Jury 8, 1991; accepted in revised&m
September 27, 1992)
Ah&act-A model of quartz pressure dissolution via grain boundary diffusion through an adsorbed water layer, incorporating the effects of silica precipitation kinetics is used to derive a simple expression indicating whether den~fi~tion by pressure dissolution is rate-limited by mass transport of p~~pi~tion kinetics. Using values of the diffisivity of thin films derived from pressure dissolution experiments, it is shown that at temperatures normally associated with pressure dissolution in sedimentary environments kinetics will be rate-limiting for most fine-gmined sandstones, and for coarser sediments, where quartz precipitation is inhibited. However, due to the strong temperature dependence of the quartz precipitation rate constant, it is demonstrated that at the elevated temperatures used in pressure dissolution experiments (typically >3OO”C) surface reactions are unimportant for particles > 2-6 pm in size. INTRODUCI’ION
occurring pressure dissolution features, no evidence for grain undercutting has been observed, and it is difficult to envisage how pressure dissolution operating by such a mechanism could give rise to the more spectacular microstructures associated with intense dissolution, such as concave-convex grain contacts and sutured grain boundaries. Ex~~menta1 studies of the compaction of halite lenses on flat silica surfaces (HICKMAN and EVANS, 199 1) found no evidence of “islandchannel” structures. Direct evidence for adsorbed water films is also elusive, although the very slow rate of transport by pure grain boundary diffusion at the temperatures normally associated with pressure solution tends to suggest that some intermediary may be present. Theoretical arguments in favour of the existence of an adsorbed water film between stressed grains is provided by the concept of disjoining pressure ( PADDAY, 1970), the equilibrium force required to remove a small increment of thickness from a thin film. TADA et al. ( 1987) estimates the disjoining pressure of a water monolayer as 270 MPa, a stress which, in a passive environment, is unlikely to be encountered at depths 5 10 km. Measurement of strain rates in pressure dissolution experiments on quartz sand led GRATIER and GUIGUET ( 1986) to conclude that pressure dissolution probably proceeded by grain boundary diffusion through an adsorbed water layer. Within most previous models of pressure dissolution, transport of material away from the site of dissolution, rather than the kinetics of the silica water system, have been assumed to be rate-limiting. RAJ ( 1982) considered the role of silica dissolution/precipitation kinetics on the rate of pressure dissolution within the “island~hannel” model. However, as this model assumes the diffusion path to be a free fluid, diffusion is relatively rapid compared to those models, which assume the diffusion path to be either an adsorbed water layer or a true grain boundary. Consequently, kinetics are significantly more important in the island channel model than in other pressure di~lution formalisms. It is the objective of this paper, using the adsorbed water layer formalism of pressure dissolution, to derive an expression to identify when quartz pressure solution may be rate-limited by precipitation kinetics and to demonstrate that such conditions may occur in nature.
WHEN A ROCK MATRIX Is subject to an applied stress, the solubility of its components will tend to increase, in accordance with thermodynamic relationships first derived by GIBBS ( 1876). In nature, one of the manifes~tions of this is the phenomena of intergranular pressure solution, first described by SORBY ( 1863). Within a porous sediment, grain boundaries normal to the lithostatic load will be more highly stressed than free faces in the pore space, the former supporting the weight of the overlying rock column, while the latter is subject only to pore fluid pressure. The increased soiubility of the rock matrix at the grain boundary, due to the excess load, will tend to drive a migration of material from the grain boundary into the surrounding pore space. At the grain scale, the processes leading to pressure dissolution are poorly understood. Models of this process fall into two broad categories. Within the first group, it is supposed that increased solubility at the grain boundary sets up a concentration gradient of the dissolved species, which results in a diffusive flow of material along the grain boundary into the surrounding pore space. This flow could occur by true grain boundary diffusion ( FARVER and YUND, 1991) by diffusion through a thin water fdm adsorbed on to the grain boundary ( WEYL, 1959; RUTTER, 1976,1983), or by diffusion through fluid channels contained between “islands” of solid in contact (RAJ and CHYUNG, 1981), possibly formed by the growth of microcracks along the grain boundary (GRATZ, 1991). Within the second group of models, increased solubility at the grain boundary is assumed to result in undercutting of the grain contact, leading to brittle failure (BATHURST, 1958) or plastic deformation ( PHARR and ASHBY, 1983; TADA et al., 1987) of the grain boundary. Numerous field and experimental studies have attempted to elucidate the undedying mechanisms behind pressure dissolution. Despite numerous petrographic studies of naturally
* Present address: The School of Materials, The University of Leeds, Leeds LS2 9JT, UK. 1499
A. M. Mullis
1500 A CRITERION FOR DETERMINING THE RATE-LIMITING MECHANISM
A mathematical description of pressure solution between truncated spherical grains of radius a, in a simple cubic packing arrangement, including the effects of silica precipitation kinetics, has been given by MULLIS ( 199 1). It may be shown that (MULLS, 199 1, Eqn. 14) if a circular grain contact of radius Q is formed between the compacting grains and the grains are separated by an adsorbed water layer of thickness ho at the grain boundary edge, in the absence of long-range transport of silica away from the site of dissolution, the concentration of dissolved silica, c’, in the pore spaces of the sediment is given by
where co is the equilibrium solubility for quartz at absolute temperature T, D is the diffusion coefficient for silica through a thin water film, 4 is the porosity of the sediment, r is the distance from the centre of the grain boundary measured in the plane of the grain contact, and k- is the local precipitation constant for quartz. This local precipitation constant for quartz is given by AIAo
k’_ = k_ G,
,_.
Equation 5 predicts that c’ will take values from co when precipitation is very rapid compared to grain boundary diffusion, up to some higher limiting value c,, given by
(
1) found that the distribution of stress across a loaded grain, and, consequently, the maximum solubility of the loaded grain boundary, is a sensitive function of the compressibility of the adsorbed water film, the central stress being greatest for the most compressible films. However, this appears to have little effect on the rate of transport of dissolved silica out from the grain boundary, as the reduction in the thickness of the film with increased compressibility reduces the width of the effective diffusion path. Consequently, we will consider only one of the solutions presented by MULLS ( I99 1 ), that in which the stress dependence of h(r), the thickness of the adsorbed water film, is such that h( r)c( Y) = constant,
(3)
1
(6)
when precipitation is very slow relative to grain boundary diffusion. Using this, we may define a normalised supersaturation .L L.I- co J‘= -, c,l? - co
(7)
which may conveniently be used to define which process is rate-limiting the pressure dissolution of quartz. For diffusion being the rate-limiting process, f- 0; and for kinetics being rate-limiting, f+ 1. Substituting for c’ and c, in Eqn. 7 and writing in terms of the composite variables, x, = g,
and
x1 = S
(8a)
(8b)
T’ we may write /‘as
ex,z(ZY-$ln($jj
where k_ is the absolute precipitation rate constant for silica, A is the wetted interfacial surface area in the system, M is the mass of water in the system, A0 = 1 m2, MO = 1 kg, and both k’ and k_ are in s-’ . MULLIS ( 199
US -RT
i ‘,,) =z CO ew
1‘=
(9) exp ;x? (
-1 1
A plot of f‘as a function of XI and X2 is given in Fig. 1. This shows that if we arbitrarily pick some value of fas indicative of the transition from diffusive to kinetic rate-limitation, the transition is approximately independent of x2; that is, it is approximately independent of the difference between the lithostatic and hydrostatic loads (in fact, X, varies by x 18% as X2 varies from values typical of 500 m burial to values typical of 5 km burial). Now, if we take j” = 0.5 as representing the transition from diffusive to kinetic rate limitation, we may determine a corresponding average value of X, for the transition. This value is X, x 0.13. That is, we may write the conditions for quartz pressure solution being rate-limited by precipitation kinetics as
whereby the solutions take on a much simpler form than is the case otherwise. Adopting this solution, we have ( MULLIS, 1991, Eqn. 15a)
hD __ > 0.13. a%_ -
( 10)
EVALUATION OF THE RATE-DETERMINING CRITERION: EXAMPLE FROM NATURE AND EXPERIMENT
where u is the molar volume of quartz, R is the gas constant, and S the applied excess stress, which is equal to Z - P, where L: is the lithostatic load and P the pore fluid pressure. By substituting for (Qdcldr) in Eqn. 1 and rearranging, we obtain
The value of Do for silica is given by WOLLAST and GARas 10e9 m2 se’ at 298 K. Above 298 K, it is assumed that diffusivity is inversely proportional to viscosity of water. The thickness of the adsorbed water layer at a grain boundary is estimated to be of the order 3 nm ( TADA et al.. 1987). The diffusion coefficient through such a water layer RELS ( 1971)
1501
Rate-limiting mechanism for quartz pressure dissolution 2.0 -
I
a5.0
-1
8.0
1.5 E 3.0
‘i Y d ,“l.O0 F \
y, 5 $ 0 z
2.0
i s Y
x”
1.0
0.5 -
f=Cb2 I
0.0
CI.6
0 I
I
I
r
0.0
1.0
0.1
0.01
X, FIG.
c’ - co
1. Values of the parameter J = -
variables X,
Gn - co
=
hD
--T-
a3k-
and
~2
=
s
7
f+
as a function of the
0 corresponds to precipitation
being very rapid compared to diffusion along the grain boundary, /1 corresponds to precipitation being very slow compared to diffusion. Approximate depths corresponding to the values of X2are given on the right-hand axis assuming that the difference between lithostatic and hydrostatic pressure is given by S = 17OOgd,where g is acceleration due to gravity and d is depth, and that the geothermal gradient is 0.035 K. km-‘, with the surface temperature being 278 K.
is likely to be less than that for bulk water, Do, although how much smaller is uncertain. By assuming that diffusivity is inversely proportional to viscosity and calculating the ap parent viscosity due to the electroviscous effect of a thin film, RUTTER ( 1976) estimates that D may be of the order of lo-l4 m* s-‘, or 10e5 Do. Laboratory experiments on the apparent viscosity of vicinal water do tend to suggest some enhancement of viscosity in thin films, with the effect being particularly noticeable in thin capillaries (CHURAYEV et al., 1970). However, the measured enhancement in surface viscosity does not seem to exceed IO- 15 times that of bulk water (PESCHEL and ADLF~NGER, 1970); furthermore, the enhancement in viscosity seems to be destroyed if a shearing force is applied to the thin film. This has tended to lead to lower values of the apparent viscosity being measured in apparatus which involve mechanically pushing two silica surfaces together, compared to static measurements, such as those made in thin capillaries. Recent experiments by HORN et al. ( 1989) and HORN and SMITH ( 1990) have found that the viscosity of water layers only a few nanometres thick was within 5% of that of bulk water, possibly for this reason. Direct measurements of diffusivity within thin water films are more difficult than viscosity measurements but have been
attempted by OLWNIK et al. (1970). They used low-energy neutron diffraction to probe the selfdiffusivity of water molecules in thin water films l-2 molecular layers thick, adsorbed onto clay minerals, and found that this quantity is only one order of magnitude less than for bulk water. One further possibility is that D may be determined directly from quartz pressure dissolution experiments. Under the assumption that pressure dissolution is rate-limited by diffusion through an adsorbed water film (which we will demonstrate in the following to be a good assumption at the elevated temperatures at which most pressure dissolution experiments are conducted), simple relationships between applied stress and strain rate may be derived (ELLIOT, 1973; RUTTER, 1976). If the applied stress in an experiment is known, the product hoD may be determined from the strain rate. This approach was used by GRATIER and GUIGUET ( 1986), who obtained the product hoD (wD, in their notation) as 2.5 X 10-‘9-2.5 X 10m2’ m3 s-’ at 360°C. They estimate from this a value forDof2.5 X 10-‘2-2.5 X 10-r3m2s-‘(=1.1 X 10-4-1.1 X lo-‘Do), although given that any adsorbed water film would have been supporting a differential stress of 150 MPa in their experiments, their estimate of ho = 100 nm would seem much too large. For a more realistic value of ho = 3 nm, we estimate that Dis in the range 8.3 X lo-“-8.3 X lo-l2 m2 s-’ (3.7 X 10m3-3.7 X 1O-4 Do). Such a large experimentally determined value for the product hoD would also appear to rule out pressure dissolution occurring by true grain boundary diffusion. FARVER and YUND ( 199 1) estimate D for a true grain boundary to be -6 orders of magnitude less than for a free fluid. R~MSTIDTand BARNES ( 1980) give the absolute precipitation constant for silica on clean quartz sand as 2598 - T,
log,, k_ = -0.707
(11)
where T is in K. For comparison with field data presented below, Thas been taken as 333 K. This gives k- = 3. I X 10m9 s-r. To facilitate a comparison between the laboratory and field data, we need to convert k- to k’_ using Eqn. 2. If we assume spherical grains of radius a in a cubic packing arrangement, we have AIAo -= M/MO
3.3 x 10-3[m] a
’
(12)
where we have taken the density of water as 1000 kgme3. Using Eqns. 10 and 12 and taking ho = 3 nm and D = 6.88 X 10-‘2-6.88 X 1O-‘3 m2 s-’ (D = 3.7 X lo-‘-3.7 X 10W4Do, where Do has been taken as 1.86 X 10e9 m2 s-’ at 333 K), we have the condition for kinetics being rate-limiting as a < 0.04-o. 12 mm; that is, kinetics are rate-limiting for fine to very fine sand particles or smaller. As the concentration of dissolved silica in the pore fluid of a sediment undergoing pressure dissolution, which is rate-limited by kinetics, can rise significantly above saturation, this effect may provide an important mechanism for the mobilisation of silica in a silt/ mud environment, from where it may be transported subsequently to adjacent coarser grained sediments. This process is described more fully by MULLIS ( 1992).
I502
A. M. Mullis
There are grounds for suspecting that in many geological settings, k’ can take values significantly different from those observed in the laboratory. BENNETT et al. ( 1988) have shown that organic acids catalyze the quartz dissolution reaction, while DOVE and CRERAR ( 1990) have shown that alkaline earth cations may also catalyze the dissolution/precipitation of quartz. Certain cations in solution, in particular A13+, may also strongly inhibit the rate of quartz precipitation, even when the amount of A13+ present constitutes less than a monomolecular layer (ILER, 1979 ). Also, observational ( HEALD, 1956, 1965) and experimental (CECIL and HEALD, I97 1) evidence shows that small amounts of clay minerals, such as illite, kaolinite, and sericite also strongly inhibit the precipitation of quartz. An in situ estimate of k’ for the Dogger Beta formation of northern Germany has been made by MULLIS ( 1991) using the data of WCHTBAUER ( 1983). By considering how far silica derived from adjacent shales had diffused in to a 14 m thick sandstone to form a bed margin cement, MULLIS ( 199 1) estimates that k’ for the sandstone is e2.95 X lo-” s-‘. The sandstone is buried to 1.6 km depth, and a temperature of 333 K was assumed for the formation of the cements. Using Eqn. 10 and taking the values of ho and D given earlier, we arrive at the condition for pressure dissolution of the sand being rate-limited by kinetics as a < 0.4-0.8 mm. That is, in the environment considered, any particle smaller than coarse to very coarse sand undergoing pressure dissolution will be rate-limited by precipitation kinetics. Unfortunately, FUCHTBALJER( 1983) does not give grain size data for the formation, so we are unable to actually say what would be the rate-limiting mechanism in this case. These conditions contrast sharply with those prevailing in most pressure dissolution experiments, where the need to obtain results in a reasonable time frame requires the use of temperatures much higher than those occurring in normal sedimentary environments. GRATIER and GUIGUET ( 1986), for example, conduct their experiments at 36O”C, which using Eqn. 1 1, would give an absolute precipitation constant of k_ = 1.54 X 10m5s-’ . Using Eqns. 10 and 12, this suggests that within their experiments, precipitation kinetics would be important only for particles 5 2-6 Frn in radius. This result is consistent with the findings of GRATIER and GUIGUET ( 1986) who, by considering the relationship between grain size and strain rate, also find that silica-water kinetics are not important in their system. CONCLUSIONS The above calculations illustrate that for values of the aqueous thin film diffusion coefficient as derived by quartz pressure dissolution experiments, silica precipitation kinetics may be an important rate-determining factor in the compaction by pressure dissolution of most fine-grained sandstones, particularly at low temperatures. Furthermore, if we accept that, in many geological settings, precipitation is likely to be slower than in the laboratory due to grain coatings and chemical inhibitors, precipitation kinetics could also be important in determining the rate of pressure dissolution over a much wider range of temperatures and grain sizes. This is in marked contrast to most laboratory experiments, where
the high temperatures normally used to obtain results within a reasonable time make it likely that precipitation kinetics would only be significant for micrometer-size particles. Thus, while pressure dissolution experiments may reproduce reasonable analogues of naturally occurring microstructure, the strain rates obtained from such experiments reflect a system in which the kinetics are fundamentally different from many natural sedimentary environments. Acknowledgments-This work was supported by NERC under grant No. GST 02/355. I am grateful to Stuart Haszeldine for help in preparation ofthe manuscript and to D. Rimstidt, R. Hellmann, and S. Brantley for their reviews of the manuscript. Editorial handling: J. 1. Drever
REFERENCES BATHURSTR. G. C. ( 1958) Diagenetic fabrics in some British Dinantian limestones. Liverpool Manchester Geol. J. 2, 1 l-34. BENNETTP. C., MELCER M. E., SIEGELD. I., and HASSETTJ. P. ( 1988) The dissolution of quartz in dilute aqueous solutions of oraanic acid at 25°C. Geochim. Cosmochim. Acta 52. 1521-1530. Cacti C. B. and HEALDM. T. ( I97 I ) Experimental investigation of the effects of grain coatings on quartz over-growth. J. Sediment. Petrol. 41, 582-584. CHURAYEVN. V., SOBOLEVV. D., and ZORIN Z. M. (1970) Measurement of viscosity of liquids in capillaries. Spec. Discuss. Faraday Sot. I,2 13-220. DOVEP. M. and CRERARD. A. ( 1990) Kinetics ofquartz dissolution in electrolyte solutions using a hydrothermal mixed flow reactor. Geochim. Cosmochim. Acta 54, 955-969. EI,LIOT D. ( 1973) Diffusion flow laws in metamorphic rocks. GSA Bull. 84,2645-2664. FARVERJ. R. and YUND R. A. ( 1991) Measurement of oxygen grain boundary diffusion in natural, fine-grained, quartz aggregates. Geochim. Cosmochim. Acta 55, 1597- 1607. FUCHTBAUERH. ( 1983) Facies control on sandstone diagenesis. In Proc NATO Advanced Study Institute on Sediment Diagenesis (ed. A. PARKERand B. W. SELLWOOD),pp. 269-288. Riedel. GIBBSJ. W. ( 1876) On the equilibrium on heterogeneous substances. Trans. Conn. Acad. 3, 108-248. GRATIER J. P. and GUIGUET R. ( 1986) Experimental pressure solution deposition on quartz grains: The crucial effect of the nature of the fluid. J. Struct. Geol. 8, 845-856. GRATZ A. J. ( 1991 ) Solution-transfer compaction in quartzites: Progress toward a rate law. Geology 19, 901-904. HEALDM. T. ( 1956) Cementation of Simpson and St. Peter sandstone in parts of Oklahoma, Arkansas, and Missouri. J. Geology64, 1630. H~EALDM. T. ( 1965) Lithification of sandstones in West Virginia. West Virginia Geol. Surv. Bull. 30, l-28. HICKMANNS. H. and EVANSB. ( I99 I ) Experimental pressure solution in halite-The effect of grain/interphase boundary structure. J. Geol. Sot. London 148,549-560. HORN R. G. and SMITH D. T. ( 1990) Measuring the surface forces to explore surface chemistry: Mica sapphire and silica. J. NonCryst. Solids 120, 72-8 I. HORN R. G., SMITH D. T., and HALLERW. (1989) Surface forces and viscosity of water measured between silica sheets. Chem. Phys. Lett. 162,404-408. ILER R. K. ( 1979) l’he Chemistry ofSilica. J. Wiley & Sons. MULLISA. M. ( 199 I ) The role ofprecipitation kinetics in determining the rate of quartz pressure solution. J. Geophys. Res. 96, 10,00710,013. MULLISA. M. ( 1992) A numerical model for porosity modification at a sandstone-mudstone boundary by quartz pressure dissolution and diffusive mass transfer. Sedimentology 39, 99-107.
Rate-limiting mechanism for quartz pressure dissolution OLEJNIK S.,STIRLING G. C., and WHITE J. W. ( 1970) Neutron scat-
tering studies of hydrated layer silicates. Spec. Discuss. Faraday Sot. 1, 194-201. PADDAY J. F. ( 1970) Cohesive properties of thin films of liquids adhering to a solid surface. Spec. Discuss. Faraday Sot. 1,64-74. PESCHELG. and ADLF~NGERK. H. ( 1970) Viscosity anomalies in liquid surface zones: IV. The apparent viscosity of water in thin layers adjacent to hydroxylated fused silica surfaces. .I. Colloid Interface Sci. 34. 505-5 10. PHARR"G. M. and ASHBY M. F. ( 1983) On creep enhanced by a liquid phase. Acta Metall. 31. 129-138. RAJ k. ( 1982) Creep in polycrystalline aggregates by matter transport through a liquid phase. J. Geophys. Res. 87,473 l-4739. RAJ R. and CHYUNG C. K. ( I98 I ) Solution precipitation creep in glass ceramics. Acta Metall. 29, 159- 166.
1503
RIMSTIDTJ. D. and BARNESH. L. ( 1980) The kinetics of silica water reactions. Geochim. Cosmochim. Acta 44, 1683-1699. RUTTER E. H. ( 1976) The kinetics of rock deformation by pressure solution. Phil. Trans. Roy. Sac. London A283, 203-209. RUTTER E. H. ( 1983) Pressure solution in nature, theory and experiment. ;I. Geol. Sot. Lond. 140,725-740. !%RBY H. C. ( 1863) On the direct correlation of mechanical and chemical forces. Proc. Roy. Sot. London 12,583-600. TADA R., MALIVAR., and SIEVERR. ( 1987) A new mechanism for pressure solution in porous quartzose sandstone. Geochim. Cosmochim. Acta 51,2295-2301. WEYLP. K. ( 1959) Pressure solution and the force of crystallizationA phenomenological theory. J. Geophys. Rex 64,2001-2025. WOLLASTR. and GARREL~ R. M. ( 197 1) Diffusion coefficient of silica in seawater. Nature Phys. Sci. 229, 94.
Gewhimica EI Cosmorhimica Acla Vol. 57: pp. 1499-I 503 Copyright 0 1993 Press Ltd. Pnnlcd in U.S.A.
Perl~arnon
Determination of the rate-limiting mechanism for quartz pressure dissolution ANDREW MARTIN MULLS* Department of Geology and Applied Geology, University of Glasgow, Lilybank Gardens, Glasgow G 12-SQQ, Scotland (Received Jury 8, 1991; accepted in revised&m
September 27, 1992)
Ah&act-A model of quartz pressure dissolution via grain boundary diffusion through an adsorbed water layer, incorporating the effects of silica precipitation kinetics is used to derive a simple expression indicating whether den~fi~tion by pressure dissolution is rate-limited by mass transport of p~~pi~tion kinetics. Using values of the diffisivity of thin films derived from pressure dissolution experiments, it is shown that at temperatures normally associated with pressure dissolution in sedimentary environments kinetics will be rate-limiting for most fine-gmined sandstones, and for coarser sediments, where quartz precipitation is inhibited. However, due to the strong temperature dependence of the quartz precipitation rate constant, it is demonstrated that at the elevated temperatures used in pressure dissolution experiments (typically >3OO”C) surface reactions are unimportant for particles > 2-6 pm in size. INTRODUCI’ION
occurring pressure dissolution features, no evidence for grain undercutting has been observed, and it is difficult to envisage how pressure dissolution operating by such a mechanism could give rise to the more spectacular microstructures associated with intense dissolution, such as concave-convex grain contacts and sutured grain boundaries. Ex~~menta1 studies of the compaction of halite lenses on flat silica surfaces (HICKMAN and EVANS, 199 1) found no evidence of “islandchannel” structures. Direct evidence for adsorbed water films is also elusive, although the very slow rate of transport by pure grain boundary diffusion at the temperatures normally associated with pressure solution tends to suggest that some intermediary may be present. Theoretical arguments in favour of the existence of an adsorbed water film between stressed grains is provided by the concept of disjoining pressure ( PADDAY, 1970), the equilibrium force required to remove a small increment of thickness from a thin film. TADA et al. ( 1987) estimates the disjoining pressure of a water monolayer as 270 MPa, a stress which, in a passive environment, is unlikely to be encountered at depths 5 10 km. Measurement of strain rates in pressure dissolution experiments on quartz sand led GRATIER and GUIGUET ( 1986) to conclude that pressure dissolution probably proceeded by grain boundary diffusion through an adsorbed water layer. Within most previous models of pressure dissolution, transport of material away from the site of dissolution, rather than the kinetics of the silica water system, have been assumed to be rate-limiting. RAJ ( 1982) considered the role of silica dissolution/precipitation kinetics on the rate of pressure dissolution within the “island~hannel” model. However, as this model assumes the diffusion path to be a free fluid, diffusion is relatively rapid compared to those models, which assume the diffusion path to be either an adsorbed water layer or a true grain boundary. Consequently, kinetics are significantly more important in the island channel model than in other pressure di~lution formalisms. It is the objective of this paper, using the adsorbed water layer formalism of pressure dissolution, to derive an expression to identify when quartz pressure solution may be rate-limited by precipitation kinetics and to demonstrate that such conditions may occur in nature.
WHEN A ROCK MATRIX Is subject to an applied stress, the solubility of its components will tend to increase, in accordance with thermodynamic relationships first derived by GIBBS ( 1876). In nature, one of the manifes~tions of this is the phenomena of intergranular pressure solution, first described by SORBY ( 1863). Within a porous sediment, grain boundaries normal to the lithostatic load will be more highly stressed than free faces in the pore space, the former supporting the weight of the overlying rock column, while the latter is subject only to pore fluid pressure. The increased soiubility of the rock matrix at the grain boundary, due to the excess load, will tend to drive a migration of material from the grain boundary into the surrounding pore space. At the grain scale, the processes leading to pressure dissolution are poorly understood. Models of this process fall into two broad categories. Within the first group, it is supposed that increased solubility at the grain boundary sets up a concentration gradient of the dissolved species, which results in a diffusive flow of material along the grain boundary into the surrounding pore space. This flow could occur by true grain boundary diffusion ( FARVER and YUND, 1991) by diffusion through a thin water fdm adsorbed on to the grain boundary ( WEYL, 1959; RUTTER, 1976,1983), or by diffusion through fluid channels contained between “islands” of solid in contact (RAJ and CHYUNG, 1981), possibly formed by the growth of microcracks along the grain boundary (GRATZ, 1991). Within the second group of models, increased solubility at the grain boundary is assumed to result in undercutting of the grain contact, leading to brittle failure (BATHURST, 1958) or plastic deformation ( PHARR and ASHBY, 1983; TADA et al., 1987) of the grain boundary. Numerous field and experimental studies have attempted to elucidate the undedying mechanisms behind pressure dissolution. Despite numerous petrographic studies of naturally
* Present address: The School of Materials, The University of Leeds, Leeds LS2 9JT, UK. 1499
A. M. Mullis
1500 A CRITERION FOR DETERMINING THE RATE-LIMITING MECHANISM
A mathematical description of pressure solution between truncated spherical grains of radius a, in a simple cubic packing arrangement, including the effects of silica precipitation kinetics, has been given by MULLIS ( 199 1). It may be shown that (MULLS, 199 1, Eqn. 14) if a circular grain contact of radius Q is formed between the compacting grains and the grains are separated by an adsorbed water layer of thickness ho at the grain boundary edge, in the absence of long-range transport of silica away from the site of dissolution, the concentration of dissolved silica, c’, in the pore spaces of the sediment is given by
where co is the equilibrium solubility for quartz at absolute temperature T, D is the diffusion coefficient for silica through a thin water film, 4 is the porosity of the sediment, r is the distance from the centre of the grain boundary measured in the plane of the grain contact, and k- is the local precipitation constant for quartz. This local precipitation constant for quartz is given by AIAo
k’_ = k_ G,
,_.
Equation 5 predicts that c’ will take values from co when precipitation is very rapid compared to grain boundary diffusion, up to some higher limiting value c,, given by
(
1) found that the distribution of stress across a loaded grain, and, consequently, the maximum solubility of the loaded grain boundary, is a sensitive function of the compressibility of the adsorbed water film, the central stress being greatest for the most compressible films. However, this appears to have little effect on the rate of transport of dissolved silica out from the grain boundary, as the reduction in the thickness of the film with increased compressibility reduces the width of the effective diffusion path. Consequently, we will consider only one of the solutions presented by MULLS ( I99 1 ), that in which the stress dependence of h(r), the thickness of the adsorbed water film, is such that h( r)c( Y) = constant,
(3)
1
(6)
when precipitation is very slow relative to grain boundary diffusion. Using this, we may define a normalised supersaturation .L L.I- co J‘= -, c,l? - co
(7)
which may conveniently be used to define which process is rate-limiting the pressure dissolution of quartz. For diffusion being the rate-limiting process, f- 0; and for kinetics being rate-limiting, f+ 1. Substituting for c’ and c, in Eqn. 7 and writing in terms of the composite variables, x, = g,
and
x1 = S
(8a)
(8b)
T’ we may write /‘as
ex,z(ZY-$ln($jj
where k_ is the absolute precipitation rate constant for silica, A is the wetted interfacial surface area in the system, M is the mass of water in the system, A0 = 1 m2, MO = 1 kg, and both k’ and k_ are in s-’ . MULLIS ( 199
US -RT
i ‘,,) =z CO ew
1‘=
(9) exp ;x? (
-1 1
A plot of f‘as a function of XI and X2 is given in Fig. 1. This shows that if we arbitrarily pick some value of fas indicative of the transition from diffusive to kinetic rate-limitation, the transition is approximately independent of x2; that is, it is approximately independent of the difference between the lithostatic and hydrostatic loads (in fact, X, varies by x 18% as X2 varies from values typical of 500 m burial to values typical of 5 km burial). Now, if we take j” = 0.5 as representing the transition from diffusive to kinetic rate limitation, we may determine a corresponding average value of X, for the transition. This value is X, x 0.13. That is, we may write the conditions for quartz pressure solution being rate-limited by precipitation kinetics as
whereby the solutions take on a much simpler form than is the case otherwise. Adopting this solution, we have ( MULLIS, 1991, Eqn. 15a)
hD __ > 0.13. a%_ -
( 10)
EVALUATION OF THE RATE-DETERMINING CRITERION: EXAMPLE FROM NATURE AND EXPERIMENT
where u is the molar volume of quartz, R is the gas constant, and S the applied excess stress, which is equal to Z - P, where L: is the lithostatic load and P the pore fluid pressure. By substituting for (Qdcldr) in Eqn. 1 and rearranging, we obtain
The value of Do for silica is given by WOLLAST and GARas 10e9 m2 se’ at 298 K. Above 298 K, it is assumed that diffusivity is inversely proportional to viscosity of water. The thickness of the adsorbed water layer at a grain boundary is estimated to be of the order 3 nm ( TADA et al.. 1987). The diffusion coefficient through such a water layer RELS ( 1971)
1501
Rate-limiting mechanism for quartz pressure dissolution 2.0 -
I
a5.0
-1
8.0
1.5 E 3.0
‘i Y d ,“l.O0 F \
y, 5 $ 0 z
2.0
i s Y
x”
1.0
0.5 -
f=Cb2 I
0.0
CI.6
0 I
I
I
r
0.0
1.0
0.1
0.01
X, FIG.
c’ - co
1. Values of the parameter J = -
variables X,
Gn - co
=
hD
--T-
a3k-
and
~2
=
s
7
f+
as a function of the
0 corresponds to precipitation
being very rapid compared to diffusion along the grain boundary, /1 corresponds to precipitation being very slow compared to diffusion. Approximate depths corresponding to the values of X2are given on the right-hand axis assuming that the difference between lithostatic and hydrostatic pressure is given by S = 17OOgd,where g is acceleration due to gravity and d is depth, and that the geothermal gradient is 0.035 K. km-‘, with the surface temperature being 278 K.
is likely to be less than that for bulk water, Do, although how much smaller is uncertain. By assuming that diffusivity is inversely proportional to viscosity and calculating the ap parent viscosity due to the electroviscous effect of a thin film, RUTTER ( 1976) estimates that D may be of the order of lo-l4 m* s-‘, or 10e5 Do. Laboratory experiments on the apparent viscosity of vicinal water do tend to suggest some enhancement of viscosity in thin films, with the effect being particularly noticeable in thin capillaries (CHURAYEV et al., 1970). However, the measured enhancement in surface viscosity does not seem to exceed IO- 15 times that of bulk water (PESCHEL and ADLF~NGER, 1970); furthermore, the enhancement in viscosity seems to be destroyed if a shearing force is applied to the thin film. This has tended to lead to lower values of the apparent viscosity being measured in apparatus which involve mechanically pushing two silica surfaces together, compared to static measurements, such as those made in thin capillaries. Recent experiments by HORN et al. ( 1989) and HORN and SMITH ( 1990) have found that the viscosity of water layers only a few nanometres thick was within 5% of that of bulk water, possibly for this reason. Direct measurements of diffusivity within thin water films are more difficult than viscosity measurements but have been
attempted by OLWNIK et al. (1970). They used low-energy neutron diffraction to probe the selfdiffusivity of water molecules in thin water films l-2 molecular layers thick, adsorbed onto clay minerals, and found that this quantity is only one order of magnitude less than for bulk water. One further possibility is that D may be determined directly from quartz pressure dissolution experiments. Under the assumption that pressure dissolution is rate-limited by diffusion through an adsorbed water film (which we will demonstrate in the following to be a good assumption at the elevated temperatures at which most pressure dissolution experiments are conducted), simple relationships between applied stress and strain rate may be derived (ELLIOT, 1973; RUTTER, 1976). If the applied stress in an experiment is known, the product hoD may be determined from the strain rate. This approach was used by GRATIER and GUIGUET ( 1986), who obtained the product hoD (wD, in their notation) as 2.5 X 10-‘9-2.5 X 10m2’ m3 s-’ at 360°C. They estimate from this a value forDof2.5 X 10-‘2-2.5 X 10-r3m2s-‘(=1.1 X 10-4-1.1 X lo-‘Do), although given that any adsorbed water film would have been supporting a differential stress of 150 MPa in their experiments, their estimate of ho = 100 nm would seem much too large. For a more realistic value of ho = 3 nm, we estimate that Dis in the range 8.3 X lo-“-8.3 X lo-l2 m2 s-’ (3.7 X 10m3-3.7 X 1O-4 Do). Such a large experimentally determined value for the product hoD would also appear to rule out pressure dissolution occurring by true grain boundary diffusion. FARVER and YUND ( 199 1) estimate D for a true grain boundary to be -6 orders of magnitude less than for a free fluid. R~MSTIDTand BARNES ( 1980) give the absolute precipitation constant for silica on clean quartz sand as 2598 - T,
log,, k_ = -0.707
(11)
where T is in K. For comparison with field data presented below, Thas been taken as 333 K. This gives k- = 3. I X 10m9 s-r. To facilitate a comparison between the laboratory and field data, we need to convert k- to k’_ using Eqn. 2. If we assume spherical grains of radius a in a cubic packing arrangement, we have AIAo -= M/MO
3.3 x 10-3[m] a
’
(12)
where we have taken the density of water as 1000 kgme3. Using Eqns. 10 and 12 and taking ho = 3 nm and D = 6.88 X 10-‘2-6.88 X 1O-‘3 m2 s-’ (D = 3.7 X lo-‘-3.7 X 10W4Do, where Do has been taken as 1.86 X 10e9 m2 s-’ at 333 K), we have the condition for kinetics being rate-limiting as a < 0.04-o. 12 mm; that is, kinetics are rate-limiting for fine to very fine sand particles or smaller. As the concentration of dissolved silica in the pore fluid of a sediment undergoing pressure dissolution, which is rate-limited by kinetics, can rise significantly above saturation, this effect may provide an important mechanism for the mobilisation of silica in a silt/ mud environment, from where it may be transported subsequently to adjacent coarser grained sediments. This process is described more fully by MULLIS ( 1992).
I502
A. M. Mullis
There are grounds for suspecting that in many geological settings, k’ can take values significantly different from those observed in the laboratory. BENNETT et al. ( 1988) have shown that organic acids catalyze the quartz dissolution reaction, while DOVE and CRERAR ( 1990) have shown that alkaline earth cations may also catalyze the dissolution/precipitation of quartz. Certain cations in solution, in particular A13+, may also strongly inhibit the rate of quartz precipitation, even when the amount of A13+ present constitutes less than a monomolecular layer (ILER, 1979 ). Also, observational ( HEALD, 1956, 1965) and experimental (CECIL and HEALD, I97 1) evidence shows that small amounts of clay minerals, such as illite, kaolinite, and sericite also strongly inhibit the precipitation of quartz. An in situ estimate of k’ for the Dogger Beta formation of northern Germany has been made by MULLIS ( 1991) using the data of WCHTBAUER ( 1983). By considering how far silica derived from adjacent shales had diffused in to a 14 m thick sandstone to form a bed margin cement, MULLIS ( 199 1) estimates that k’ for the sandstone is e2.95 X lo-” s-‘. The sandstone is buried to 1.6 km depth, and a temperature of 333 K was assumed for the formation of the cements. Using Eqn. 10 and taking the values of ho and D given earlier, we arrive at the condition for pressure dissolution of the sand being rate-limited by kinetics as a < 0.4-0.8 mm. That is, in the environment considered, any particle smaller than coarse to very coarse sand undergoing pressure dissolution will be rate-limited by precipitation kinetics. Unfortunately, FUCHTBALJER( 1983) does not give grain size data for the formation, so we are unable to actually say what would be the rate-limiting mechanism in this case. These conditions contrast sharply with those prevailing in most pressure dissolution experiments, where the need to obtain results in a reasonable time frame requires the use of temperatures much higher than those occurring in normal sedimentary environments. GRATIER and GUIGUET ( 1986), for example, conduct their experiments at 36O”C, which using Eqn. 1 1, would give an absolute precipitation constant of k_ = 1.54 X 10m5s-’ . Using Eqns. 10 and 12, this suggests that within their experiments, precipitation kinetics would be important only for particles 5 2-6 Frn in radius. This result is consistent with the findings of GRATIER and GUIGUET ( 1986) who, by considering the relationship between grain size and strain rate, also find that silica-water kinetics are not important in their system. CONCLUSIONS The above calculations illustrate that for values of the aqueous thin film diffusion coefficient as derived by quartz pressure dissolution experiments, silica precipitation kinetics may be an important rate-determining factor in the compaction by pressure dissolution of most fine-grained sandstones, particularly at low temperatures. Furthermore, if we accept that, in many geological settings, precipitation is likely to be slower than in the laboratory due to grain coatings and chemical inhibitors, precipitation kinetics could also be important in determining the rate of pressure dissolution over a much wider range of temperatures and grain sizes. This is in marked contrast to most laboratory experiments, where
the high temperatures normally used to obtain results within a reasonable time make it likely that precipitation kinetics would only be significant for micrometer-size particles. Thus, while pressure dissolution experiments may reproduce reasonable analogues of naturally occurring microstructure, the strain rates obtained from such experiments reflect a system in which the kinetics are fundamentally different from many natural sedimentary environments. Acknowledgments-This work was supported by NERC under grant No. GST 02/355. I am grateful to Stuart Haszeldine for help in preparation ofthe manuscript and to D. Rimstidt, R. Hellmann, and S. Brantley for their reviews of the manuscript. Editorial handling: J. 1. Drever
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