Journal of Structural Geology 22 (2000) 1395±1407 www.elsevier.nl/locate/jstrugeo
Kinetics of crack-sealing, intergranular pressure solution, and compaction around active faults
Franc° ois Renard a,*, Jean-Pierre Gratier b, Bjùrn Jamtveit a
aFluid±Rock Interactions Group, University of Oslo, Departments of Geology and Physics, P.O. Box 1047 Blindern, 0316 Oslo, Norway bLGIT, CNRS-Observatoire, Universite Joseph Fourier BP 53, 38041 Grenoble cedex, France Received 17 January 2000; accepted 18 May 2000
Abstract
Geological evidence indicates that ¯uids play a key role during the seismic cycle. After an earthquake, fractures are open in the fault and in the surroundings rocks. With time, during the interseismic period, the permeability of the fault and the country rocks tends to decrease by gouge compaction and fracture healing and sealing. Dissolution along stylolite seams provides the matter that ®lls the fractures, whereas intergranular pressure solution is responsible for gouge compaction. If these processes are fast enough during the seismic cycle, they can modify the creep properties of the fault. Based on ®eld observations and experimental data, we model the porosity decrease by pressure solution processes around an active fault after an earthquake. We arrive at plausible rates of fracture sealing that are comparable to the recurrence time for earthquakes. We also study the sensitivity of these rates to various parameters such as grain size, fracture spacing, and the coecient of diusion along grain boundaries and stylolites. 7 2000 Elsevier Science Ltd. All rights reserved.
1. Roles of ¯uids in faults zone (Rice, 1992). On the other hand, high overpres- sure can be sustained by conjecturing eciently sealed The interest in slip instability has increased consider- compartments and gouge compaction (Byerlee, 1990; ably since Brace and Byerlee (1966) proposed that it Sleep and Blanpied, 1994). In both cases, earthquakes might be related to earthquake rupture (see Scholz, are assumed to take place when the ¯uid pressure is 1990 for an historical review). As geologic evidence is close to lithostatic values. Because of dilatancy and gathered on the involvement of ¯uids in faults, a grow- fault connectivity increase, earthquakes are assumed to ing number of researches have started to study the reduce the ¯uid pressure to hydrostatic levels. As a eects of ¯uid pressure on fault mechanics (Lachen- consequence, the ¯uid pressure varies between two bruch, 1980; Sleep and Blanpied, 1994; Chester, 1995; end-member values during the seismic cycle. Hickman et al., 1995; Segall and Rice, 1995; Matthai In a model of ¯uid ¯ow in partially sealed fault and Fisher, 1996; Lockner and Byerlee, 1995; Four- zones, Sleep and Blanpied (1994) assumed a purely vis- nier, 1996; Miller et al., 1996; Yamashita, 1997; Hen- cous rock behavior. The pressure equation was based derson and Maillot, 1997). Byerlee (1990) and Rice on the conservation of ¯uid phase and included both (1992) suggested that the relative weakness of some an increase in pressure due to compaction and a faults could be due to high intra-fault ¯uid pressure. decrease in pressure due to escape of ¯uid. In the case Overpressuring could be generated and maintained by of a single fault system, they showed that the increased the ¯ow of deep ¯uids into the ductile roots of a fault ¯uid pressure allows frictional failure and earthquakes at shear traction far below that required when ¯uid pressure is hydrostatic. Segall and Rice (1995) have * Corresponding author. developed a model of unstable slip by modifying the E-mail address: [email protected] (F.Fran. Renard). rate and state dependent friction model to include ¯uid
0191-8141/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S0191-8141(00)00064-X 1396 F. Renard et al. / Journal of Structural Geology 22 (2000) 1395±1407 pressure and porosity. This model is based on porosity against solution surface) and by chemical arguments changes deduced from gouge experiments (Marone et (same mineral dissolved in solution cleavage and preci- al., 1990) with overcompacted rocks where one ob- pitated in veinsÐsee Fig. 1) (Gratier et al., 1994). serves an increase of porosity with shear strain if the Therefore pressure solution associated with deposition normal stress is kept constant. This is called the dila- in veins is the main mechanism of crack sealing tancy hardening eect (Rice, 1975; Rudnicki and around active faults and in the gouge. Chen, 1988). Henderson and Maillot (1997) used an The nature and the origin of ¯uids and sealing min- empirical exponential function to describe the re- erals in fault systems can be estimated from isotope duction of porosity due to compaction. They assumed studies. In the San Andreas system, California, the iso- that in the case of rupture, porosity could increase up tope composition of samples from gouges and vein-®ll- to 10%. Miller et al. (1996) presented a discrete ing minerals have been compared with that of the host instability model assuming that the normal stress com- rocks by Pili et al. (1998). The results indicate that pacts the fault zone and reduces the porosity, resulting fault zones formed at depths between 2 and 5 km in high ¯uid pressured compartments and a local appear to have been in®ltrated during deformation by weakening of the fault. ¯uids of deep origin dominated by crustal water. Min- As this short literature survey reveals, porosity and eral deposit in gouges and veins may thus be con- permeability are assumed to vary during the seismic sidered as representing a mixture of species dissolved cycle, but the porosity changing mechanisms and the in the external ¯uid and species from the dissolving precise time scales of these mechanisms are still contro- country rocks. versial. This is compatible with observations indicating that In addition to these models, observations of from time to time, probably after an earthquake occur- exhumed faults provide many evidence of extensive rence, gouge and veins are hydraulically connected. ¯uid±rock interactions (Evans and Chester, 1995), in Then, during the interseismic period, vein sealing particular, pressure solution processes can be demon- occurs by input of material removed by pressure sol- strated by indented grains and stylolites (Gratier and ution processes in the nearby country rocks. This type Gamond, 1990; Janssen et al., 1997). Some of the most of crack seal model is also compatible with the results signi®cant observations may be summarized as follows. of Fisher et al. (1995) who found that there is no evi- Fluid inclusion networks in the vein minerals demon- dence for long distance transport of silica within the strate the presence of ¯uids during the crack sealing Kodiak convergent prism in Alaska. However, there is and re¯ect self-healing processes within the sealing geochemical evidence of local migration of silica from deposit. However, the veins related to active faults are the rock matrix to veins with an approximate balance always sealed by external input of material (Gratier et between the silica dissolved from the rock matrix and al., 1994; Evans and Chester, 1995). The source of this the amount of quartz precipitated in veins (Fisher and vein material is mostly nearby solution cleavage and Brantley, 1992). stylolites in the country rocks. This can be demon- From the observed natural structures, the following strated both by geometric arguments (veins interrupted sequence of crack-sealing processes is expected to occur in active fault zones during interseismic periods (Gratier et al., 1994). Initially, rapid slip driving an earthquake will cause an increase of the overall per- meability and reduce the ¯uid pressure to levels approaching hydrostatic values within the fault zone. The ®rst stage of ¯uid/rock interactions occurs on the free faces produced during the earthquake and is characterized by fast kinetics. Self-healing of fractured minerals and growth of metamorphic minerals are rel- evant to this stage. However, none of these processes signi®cantly contribute to the sealing of the open cracks (Gratier et al., 1994). The second stage is press- ure solution creep, associated with the indentation of grains and the dissolution along stylolites. It is found to be much more ecient and can be thought as a mechanism that relaxes the stresses in and around the Fig. 1. Thin section of crack sealing of veins in a granite rock from fault. It also decreases the eective stress imposed on the San Gabriel Fault, California. Dissolution occurs along stylolites (S) and solutes precipitate locally in veins open in tension (T ). In the matrix by compaction of the gouge. According to this case, the cracks are thin (5±10 mm) and the mean distance experiments, the time scales of such crack sealing, con- between stylolites is 100±200 mm. Adapted from Gratier et al. (1994). trolled by the kinetics of pressure solution and associ- F. Renard et al. / Journal of Structural Geology 22 (2000) 1395±1407 1397
gouge compaction by intergranular pressure solution
L z decreases with time
L z
L x
pore free face L z contact: area A c
grain radius: Lf a) compaction by pervasive pressure solution at grain scale
crack sealing by pressure solution along stylollites
L sz decreases with time stylolite
L sz
stylolite stylolite
cementation in the vein open vein s dissolution along the stylolite (thickness L ) 1 v stylolite
vein
L sz
L sy L sx stylolite b) compaction within fractured rocks by pressure solution along stylolites and vein cementation
Fig. 2. Textural model for compaction by pressure solution during the interseismic period. Compaction occurs through two mechanisms: perva- sive pressure solution at the grain scale (gouge), and vein cementation associated with dissolution along stylolites (country rock around an active fault). (a) Model of truncated sphere for intergranular pressure solution. (b) Model of vein cementation and dissolution along stylolites. The dierent lengths are used in Eqs. (2)±(10). Both mechanisms are responsible of a creep-like deformation of the rock, with dierent characteristic time scales. 1398 F. Renard et al. / Journal of Structural Geology 22 (2000) 1395±1407 ated with deposition processes, are on the order of sev- ameters such as stress, rock texture, diusion rates and eral tens of years to several million years and are temperature are also studied. strongly dependent on temperature and the rock tex- ture (Rutter, 1976; Hickman and Evans, 1991; Gratier, 1993). Modeling such an evolution requires knowledge 2. A model for pressure solution and crack sealing of the geometry of the transport pathways controlling the rate of mass transfer. In this paper, we consider 2.1. The driving force for pressure solution two schemes of porosity reduction 1. Compaction modeling of the gouges by pervasive Stress has a signi®cant eect on the chemical poten- pressure solution at grain contacts associated with tial of a solid. The eect is to increase its molar free precipitation of the dissolved matter in the pore energy compared to that at zero-stress. Though the space (Fig. 2a). This mechanism leads to grains Gibbs-type free energy cannot be de®ned for nonhy- indentation and an overall porosity decrease drostatically stressed solids (Shimizu, 1995), dissol- (Renard et al., 1999). ution/precipitation of solids is fully described by the 2. Crack-seal and creep around the faults by mass surface chemical potential m (Gibbs, 1877). If tangen- transfer between dissolution surfaces (i.e. stylolites), tial stress on the grain surface is zero, the chemical po- oriented perpendicularly to the maximum compres- tential dierence over a grain surface can be written sive stress direction and associated with precipi- tation on surfaces in contact with free ¯uid in open Dm Df VsDPn 1 cracks (Fig. 2b). where m is the chemical potential of the dissolved com-
We attempt to quantify the kinetics of these two ponent, Pn is the normal pressure on the solid, f rep- mechanisms by a conceptually simple reaction and resents the molar Helmholtz free energy, Vs is the local transport model. The eects of various par- molar volume of the solid (see Table 1 for notations
Table 1 Symbols used in Eqs. (1)±(19) and Fig. 2
F Free energy (J mole 1) m Chemical potential (J mole 1) m0 Chemical potential in a reference state Keq Equilibrium constant of a chemical reaction K0 Equilibrium constant in a reference state c Concentration of silica in solution (mole m 3) H4SiO4 a Activity of silica in solution H4SiO4 g Activity coecient of silica in solution H4SiO4 3 cc Concentration within the grain contact (mole m ) 3 cs Concentration in the stylolite (mole m ) 3 cp Concentration in the pore ¯uid (mole m ) cv Concentration in vein 2 1 Kp Kinetics constant for precipitation of the pore surface (mole m s ) D Thickness of the water ®lm trapped inside the grain contact (m) 2 1 Dc Coecient of diusion along the grain contact (m s ) 3 1 Vs Molar volume of a mineral (m mole ) T Temperature (K)
PI Normal stress in direction i (x, y or z ) (bar) Ps Normal stress to a stylolite (bar) Pp Pore pressure (bar) f Gouge porosity ffrac Fracture porosity Lx,Ly,Lz Grain parameters to describe the truncated sphere geometry of Fig. 2(a) (m) Lv Thickness of the open vein (m) Lsx,Lsz Distance between stylolites in the x and y directions (m) Lf Grain radius (m) 2 Ac Surface area of contact between two grains (m ) 2 lc Radius of a grain contact (plc=Ac) (m) 1 Gc Velocity of dissolution at grain contacts (m s ) 1 Gp Velocity of precipitation on the pore surface (m s ) 1 Gs Velocity of dissolution along a stylolite (m s ) 1 Gv Velocity of precipitation in the open vein (m s ) F. Renard et al. / Journal of Structural Geology 22 (2000) 1395±1407 1399 and units). Relationship Eq. (1) characterizes the where Ac is the surface area of the contact and Lx, Ly chemical potential at the solid at each point on the are the truncations of the spherical grains in the x and surface of the solid and varies over the grain surface y directions, as shown in Fig. 2(a). As Ac increases between contact areas and pore surface. The driving with time, the stress normal to the contact decreases force for material transfer along a grain surface is the allowing a coupling between grain geometry and stress. dierence in chemical potential between two parts of For simplicity, we focus on the vertically directed the same crystal surface. The term Df contains contri- grain contacts (normal to the z-axis). Results are iden- butions due to elastic energy, dislocation energy and tical for the horizontally directed contacts (normal to surface energy. At the conditions prevailing in the x and y contacts) because one considers an isotropic earth's crust, these dierent contributions are one to loading of the system. 1 two orders of magnitude less than the contribution of The velocities Gc and Gp (m s ) describe the geo- the normal stress, and this is often neglected (Paterson, metric evolution of the z-contact length variables. Gc is 1973). representative of the time evolution of the moving grain interface due to dissolution at contact and rep-
2.2. A textural and stress model for intergranular resents the rate at which Lz decreases with time. Gp pressure solution takes in account the evolution of the grain radius and characterizes the thickness of overgrowth on the pore The non-linear dynamics of grain compaction in the surface (Renard et al., 1999). Given these rates, the gouge arise from the fact that during deformation textural evolution of the grains follows: intergranular contact areas grow and grains indent dLz dLf Á each other. To account for such geometric evolution, Gc cc,Pc,Pz and Gp cp,Pp,Pz 3 the rock is modeled as a cubic array of grains (Dewers dt dt and Ortoleva, 1990; Renard et al., 1999) and assumed where Lf is the grain radius, Lz is the length of the to be monomineralic. The grains have a truncated truncation of the spherical grain, cc and cp are the con- spherical geometry (Fig. 2a), which is characterized by centrations inside the contact and in the pore respect- four geometrical lengths: Lf, the grain radius, and Lx, ively; Pc is the normal stress in the contact, Pp is the Ly, and Lz, the lengths of the truncations in the three pore pressure and Pz is the lithostatic vertical pressure. directions of space. Intergranular ¯uids can be con- The rates G are de®ned positive when dissolution sidered as ¯uid ®lms (Weyl, 1959; Rutter, 1976) or as occurs on the surface and negative for precipitation. In a connected network of ¯uid inclusions, forming an the model cc is related to cp by a steady state assump- `island and channel' pattern (Raj and Chyung, 1981; tion (Dewers and Ortoleva, 1990). Fick's Law gives Spiers et al., 1990). In our simulations we will simply describe the grain boundary as a ¯uid ®lm. This 2plcDDc cc cp Ac assumption should only modify the transport proper- Gc 4 lc Vs ties of the grain contacts, the diusion being faster in the presence of an `island and channels' structure. where Vs is the molar volume of the dissolving solid, Compaction at a grain scale can be divided into Dc is the diusion coecient along the grain bound- three successive steps (Raj, 1982; Renard et al., 1997). aries, D is the thickness of the water ®lm trapped in Dissolution occurs at grain contacts due to a local the grain interface, lc is the radius of the circular grain concentration of stress. Then solutes diuse along the contact, and Ac its surface area. We must also take grain/grain interface where a ¯uid ®lm is trapped into account the conservation of volume in a system (Renard and Ortoleva, 1997). When arriving in the closed at a grain scale pore, the solutes precipitate on the surface of the grain in contact with the pore space. The whole process is 6AcGc ApGp 0: 5 responsible for a non-linear porosity reduction with The factor 6 in Eq. (5) represents the six contacts that time. each grain has with its neighbors. The rate Gp corre- The complex dynamics of pressure solution comes sponds to the precipitation on the pore surface, which from a non-linear feedback between stress and rock is equal to texture. Following the analysis by Dewers and Orto- leva (1990) one can write a force balance relationship cp Gp kpVs 1 6 between the vertical stress Pz (equivalent to a litho- Keq static stress, for example the eect of overburden), the normal stress on a grain contact Pc, and the pore where kp is the kinetics constant for mineral precipi- pressure Pp: tation and Keq the equilibrium constant for the reac- Á Á tion of dissolution/precipitation. The ratio in Eq. (6) LxLy Pz Pp Pz AcPc LxLy Ac Pp 2 represents the degree of super-saturation. This model 1400 F. Renard et al. / Journal of Structural Geology 22 (2000) 1395±1407 of dissolution at grain contacts and precipitation on Ds is the thickness of the stylolite, and Ds is the diu- the free face allows an overall compaction of the rock. sion coecient of the dissolving solute along the stylo- lite. The value of the diusion coecient along a stylolite is not well de®ned, but for a given tempera- 2.3. A textural and stress model for stylolites and veins ture, it should range between that at a single grain contact (as measured in pressure solution experiments) Dissolution occurs along the stylolite planes and and that in a very ®ne-grained rock such as an ultra- matter is transported to the open veins oriented per- mylonite or a quartz aggregate (Farver and Yund, pendicularly to the stylolites. The mean distance 1995, 1999). For example, at 2008C, a typical values between stylolites L decreases with time due to dissol- sz for the diusion coecient in a mylonite, estimated ution. During the same time, the open veins close after Farver and Yund (1999), is 9.8 Â 10 15 m2 s 1, because of precipitation and L and L are increasing sx sy with an activation energy of 30 2 6 kJ mole 1. This whereas the open space in veins L is decreasing v value can vary within an order of magnitude, depend- (Fig. 2b). The process stops when the original open ing on whether diusion is parallel or perpendicular to fracture becomes ®lled and attains zero percent poros- the mylonite schistosity. Pressure solution experiments ity. The fracture porosity f of the system is equal frac indicate that the product of water ®lm thickness and to coecient of diusion is around 10 19±10 20 m3 s 1 at Á L L L 3508C (Rutter, 1976; Gratier and Guiguet, 1986) with f v sx sy Á: 7 frac L L L L a smaller activation energy. For a ¯at grain interface sx v sy v of 1 nm, this means that the coecient of diusion at As in the case of intergranular pressure solution, the the grain contact would be 104±105 greater than for stress normal to the stylolite surface decreases with the case of an ultramylonite. time as precipitation occurs in the veins. One can also As for the model of pressure solution at grain scale, write a relationship similar to Eq. (2) between the we must take into account the conservation of volume in a closed system. All the matter dissolved inside the macroscopic stress perpendicular to the stylolite Pz, (here similar to a lithostatic stress), the normal stress stylolite precipitates in the veins. on a stylolite surface Ps, and the pore pressure Pp 2 2LsxGs 4LsxLszGv 0: 11 2 2 Lsx Lv Pz LsxPs 2LsxLvPp 8 The rate Gv, representing precipitation on the vein surface, is de®ned as in Eq. (6), except that the concen- where it is assumed that Lsx=Lsy. Deformation occurs by dissolution of matter in sty- tration is that in the open vein lolite planes perpendicular to the main stress s and 1 cv solutes precipitate in open veins perpendicular to Gv kpVs 1 : 12 Keq s2=s3. Thus it follows that dL sz G c ,P dt s s s 9 3. Modeling the compaction within and around a fault dL dL dL Á and sx sy v G c ,P dt dt dt v v p The gouge is considered to be an aggregate of either where L and L are assumed to be equal. The rate pure quartz or pure calcite with a grain size of 10 mm sx sy in the gouge. Just after an earthquake, the porosity of Gs describes the decrease of thickness of the interstylo- litic region due to dissolution along the stylolite sur- this rock is high and estimated to be around 10% face; c is the concentration of dissolved matter in the from evaluation of the mean ratio between sealed s cracks and host rocks. This value is also taken as an stylolite and Ps the stress normal to the stylolite. Gv is the rate of precipitation inside the vein, c is the con- initial condition for gouge compaction to compare the v two processes. The initial mean distance between veins centration of solutes in the open vein and Pp the ¯uid pressure inside the vein, assumed to be equal to the and stylolites in the country rock is taken from geo- pore pressure (equal to s and s ). logical observations (Lsx=Lsy=Lsz=100 mm) (Fig. 1; 2 3 Gratier et al., 1994; Gudmundsson, 1999). Just after In the model cs is related to cv by a Fick's law describing diusion from the stylolite to the veins an earthquake, the fractures are considered to be open, with a thickness Lv=6 mm (Gratier et al., 1994). 2 Given these conditions, the initial fracture porosity of 4LsyDsDs cs cv Lsx Gs 10 the system is close to 10%. We apply our model to Lsy Vs estimate pressure solution rates in pure limestone and where Vs is the molar volume of the dissolving solid, quartz rocks. These initial conditions will be varied F. Renard et al. / Journal of Structural Geology 22 (2000) 1395±1407 1401 later to study the eect of the distance of transport ution. At equilibrium, if we take the reference states of between sites of dissolution and sites of precipitation. solid quartz and water to be pure stress-free states and The transport properties of the grain interface and we take the activities of quartz and water equal to 1. the stylolites are taken from experiments on pressure The equilibrium constant is related to silica solubility solution (Rutter, 1976; Hickman and Evans, 1991; c through H4SiO4 Gratier and Guiguet, 1986; Gratier, 1993) which give K g c 15 the product coecient of diusion times water ®lm eq H4SiO4 H4SiO4 thickness in the range 1019±1021 m3 s 1 at 250 to where g is the activity coecient, assumed to be 3508C. A small activation energy of 15 kJ/mole is H4SiO4 equal to 1 in a dilute pressure free state and c is taken to estimate this product at lower temperatures. H4SiO4 the molarity of aqueous silica. K under stress is re- The thickness of the ¯at grain interfaces is assumed to eq lated to the equilibrium constant K in stress-free state decrease exponentially as the eective stress increases 0 by (Renard and Ortoleva, 1997). We consider now two simple end-member cases. The m m K K exp qz 0 : 16 rock is monomineralic and can be either pure quartz eq 0 RT or pure calcium carbonate. The chemical reaction of dissolution or precipitation of quartz is taken to be Using these relations one can estimate the solubility of quartz on the dierent sites of the model (grain con- SiO2 s2H2O , H4SiO4 aq: 13 tact, pore surface, stylolites, veins). Such solubility enters the model through the reaction of precipitation The equilibrium constant for this reaction Keq is in the pore (Eqs. (6) and (12)). In Eq. (16), K0 is a almost only dependent on temperature, the pressure K H4SiO4 14 eq a a dependence being included in the free-energy term. It SiO2 H2O is the equilibrium constant for a stress-, dislocation-, where all the as are the activity of the species in sol- elastic- and surface energy-`free' state. We choose its value to be as following (Rimstidt and Barnes, 1980)
10 1560 log K 1:881 0:002028 Â T 17 0 T
9 km 8 km 7 km 6 km 5 km 4 km 3 km 2 km The rate constant for dissolution/precipitation increases with temperature and varies with pH and ionic concentrations (Dove, 1994). Some authors have evaluated the kinetics of quartz dissolution through laboratory measurements (Dove, 1994) and found rates faster than in geological conditions (Oelkers et 5 al., 1996). Walderhaug (1994) has derived precipitation rates in sandstones in the Norwegian shelf from ¯uid inclusion microthermometry and he has found a value much lower. Because of the large uncertainty about the kinetics of quartz dissolution-precipitation, we have chosen kinetics laws derived from data measured
Gouge porosity in a quartz-rich rock (%) in laboratory (Dove, 1994), even if in geological con- ditions inhibitions processes could slow down the kin- grain size: 10 microns etics of dissolution-precipitation. 0 The reaction of calcite dissolution is taken to be 10-5 10-3 10-1 101 2 2 Time (years) CaCO3 s,Ca CO3 : 18 Fig. 3. Simulations of the compaction of a quartz gouge with ®ne Plummer and Busenberg (1982) give the following re- grains (10 mm). The porosity decreases with time due to pervasive lationship for the equilibrium constant of this reaction pressure solution. Initially, the porosity of the gouge is high (around between 0 and 908C 10%) and it decreases because of dissolution at grain contacts and Á precipitation in the pores (see Fig. 2a). The porosity±time curves are log K 171:065 0:0077993T almost linear in this logarithmic±linear graph indicating an exponen- eq tial decrease of porosity with time with characteristic times decreas- ing with depth. In these and the next simulations, the temperature 2839:319=T 71:595 log T 19 gradient is 308C/km and the lithostatic and hydrostatic pressure gra- dients are taken to be 22 MPa/km and 10 MPa/km, respectively. where T is the temperature in Kelvin. We have also 1402 F. Renard et al. / Journal of Structural Geology 22 (2000) 1395±1407 used their model of CO2±H2O equilibrium to estimate evolution of the porosity of both the gouge and the 2 the concentrations of CO3 ,HCO3 , and H2CO3,as country rocks, but also on the ¯ux of ¯uid from depth well as the activity of dissolved. CO2 between 0 and (Rice, 1992). As we intend to compare the rate of 908C. Above this temperature, the equilibrium con- change of porosity between two mechanisms (grain stant for this reaction is estimated with the compaction, and stylolites±veins) we think it is inter- SUPCRT92 database (Johnson et al., 1992). The kin- esting to use a comparable driving force. However it etics constant of calcite dissolution and precipitation must be kept in mind that the evolution with time of was calculated from Plummer et al. (1978, 1979). both rates of change of porosity may be slower than Compaction and stylolitization are modeled for estimated when overpressuring develops. dierent geological conditions corresponding to a ver- For stylolites and vein-sealing the driving force is tical pro®le in the crust along a fault between 2 and 9 the shear along the fault plane which induces a dier- km depth. A temperature gradient of 308 km 1 is ence of normal stress between the stylolites and the taken. As stated in the introduction, the driving force veins. Few data are available on the evolution of such for gouge compaction after an earthquake is assumed a dierence of stress with depth; so we took an evol- to be the dierence between lithostatic and hydrostatic ution similar to the driving force for the gouge, assum- pressure. The hydrostatic pressure gradient is chosen ing a constant increase of the normal stress dierence to be 10 MPa km 1 and the lithostatic pressure gradi- with depth (12 MPa km 1). Moreover it has been ent is 22 MPa km 1 (here assumed to correspond to observed that in the upper crust the eective stress s1). We will consider only the case of a hydrostatic variation has a smaller in¯uence on the rate of defor- ¯uid pressure. Overpressure compartments may mation, than the temperature variation (Oelkers et al., develop in the faults during the interseismic period, 1996; Walderhaug, 1994). In our simulations, the and this should modify the driving force for pressure increase of temperature with depth induces a greater solution deformation. However, overpressuring is a eect on the increase of the pressure solution rate than complex process which is dependent not only on the eective stress variations would.
a) b) crack thickness: 6 microns crack thickness: 6 microns disntance between cracks: 100 microns distance between cracks: 100 microns 10 10
9 km 8 km 7 km 6 km 5 km 4 km 3 km 2 km
9 km
8 km
5 5 7 km 2 km 3 km 4 km 5 km 1 km 6 km 2 km 6 km 7 km 3 km 8 km 4 km 9 km 5 km
Fracture porosity in a carbonate rock (%) 6 km
Fracture porosity in a quartz-rich rock (%) 7 km 8 km 9 km
0 2 3 4 5 0 10 10 10 10 101 102 103 104 Time (years) Time (years)
Fig. 4. (a) Pressure solution by stylolitization and precipitation in quartz veins as a function of time. The fracture porosity, initially arbitrarily ®xed at around 10% just after an earthquake, decreases with time. The curves correspond to dierent depths between 2 and 9 km. (b) Pressure solution by stylolitization and precipitation in calcite veins as a function of time. As for quartz, the fracture porosity is arbitrarily ®xed at around 10% just after an earthquake. The curves correspond to simulations at dierent depths between 1 and 9 km. Between 2 and 5 km, the rate of pressure solution does not change with depth. This is due to two antagonistic processes. On one hand, the solubility of calcium carbonate decreases with temperature and depth, lowering the pressure solution rate. On the other hand the increasing eective stress with depth promotes the rate of pressure solution. These two eects cancel with depth. Below 5 km, the solubility decrease is faster that the increase of driving force related to stress. Therefore the rate of pressure solution decreases. The characteristic times for deformation are greater for crack sealing than in the gouge (Fig. 3), mainly because of an increase of the distance of transport. F. Renard et al. / Journal of Structural Geology 22 (2000) 1395±1407 1403
3.1. Quartz grain deformation in the gouge rock decreases with time (Fig. 4a). If this porosity is arbitrarily ®xed at 10% after an earthquake, it During chemical compaction, dissolution occurs decreases to 0% in 1000 y at 9 km depth, or 0.4 My at inside the grain contact and matter precipitates on the 2 km depth for a quartz-rich lithology. These fracture- pore surface, hence Lf (measuring the thickness of healing rates are 3±4 orders of magnitude slower than overgrowth on the pore surface) increases (Fig. 2a). It for compaction of the gouge. They correspond to de- is assumed that chemical compaction is isotropic, so formation rate increasing by two orders of magnitude grain truncation lengths Lx, Ly, and Lz all decrease from 2 to 9 km (Fig. 5). equally with deformation. The model is closed at the In the case of calcite, the fracture porosity closes in grain scale with respect to solid mass, thus the grain about 1000 y (Fig. 4b) between 2 and 5 km and the volume does not change with time. The grains evolve compaction rate slows down below. It seems that the as truncated spheres with decreasing pore surface area rate of pressure solution does not change much with and increasing nominal contact area (Fig. 2a). Simul- depth in carbonate-rich rocks between 2 and 5 km. taneously, porosity decreases with time (Fig. 3). This is due to two opposite eects that cancel each Depending on depth, it takes from less than 1 day (at other. On one hand, an increase of temperature with 9 km) to 50 y (at 2 km) to close the porosity in a depth decreases calcite solubility and the rate of press- quartz-rich gouge with very ®ne texture (grain size of ure solution. On the other hand, the eective stress 10 mm). In this case of very ®ne-grained rocks, press- increases with depth and promotes the driving force ure solution would be very fast and it would represent for pressure solution. Below 5 km, the rate of pressure a relevant mechanism for post-seismic viscous relax- solution decreases constantly because the solubility ation in the gouge of an active fault at depth. decrease is not compensated by the increase of eective stress. 3.2. Stylolite±vein deformation in the fault
Due to quartz dissolution along stylolites and pre- cipitation in the veins, the fracture porosity ffrac of the
9 km 8 km -12 10 7 km Dp Dp Da Da Dgb Dgb
10 6 km 5 km
10-13 4 km 2 km
3 km 5
-14 10 2 km Deformation rate (s-1) for veins in a quartz rock
102 103 104 105 Fracture porosity in a quartz-rich rock (%) Time (years) 5 km 2 km5 km 2 km 5 km 0 Fig. 6. Sensitivity of the rate of crack sealing to the value of the 10-1 101 103 105 107 109 coecient of diusion along the stylolite for simulations correspond- Time (years) ing to a quartz-rich rock at 2 and 5 km depth. Dgb is the coecient of diusion as estimated by Farver and Yund (1999) for a highly 1 Fig. 5. Deformation rates for crack sealing in a quartz-rich rock. In- stressed mylonite (activation energy 30 kJ mole ), Da is the coe- itial conditions are the following: mean distance between stylolites cient of diusion for a 2-nm-thick water ®lm as estimated from 1 (Lsz): 100 mm, crack aperture (Lv): 6 mm. The deformation rate pressure solution experiments (activation energy 15 kJ mole ) (Rut- decreases slowly with time as deformation occurs because the stress ter, 1976; Hickman and Evans, 1991; Gratier and Guiguet, 1986; normal to stylolites decreases as cracks seal (Eq. (8)). The eect of Gratier, 1993) and Dp is the coecient of diusion in bulk water (ac- depth on the rate of pressure solution is clearly seen, the main eect tivation energy 15 kJ mole 1) (Applin, 1987). The time associated being the increase of temperature that modify the solubility of silica with pressure solution is strongly dependent on the diusion rate at in water by several orders of magnitude (Eqs. (9) and (10)). grain boundaries. 1404 F. Renard et al. / Journal of Structural Geology 22 (2000) 1395±1407
a) b) distance between cracks (micron) distance between cracks (micron) 10 20 50 100 200 500 1000 50 100 200 500 1000
0.1 10
0.05 5 Fracture porosity in a carbonate-rich rock (%) Fracture porosity at 5 km in a quartz-rich rock
0 0 0 1 2 3 4 5 6 101 103 105 107 10 10 10 10 10 10 10 Time (years) Time (years)
Fig. 7. (a) Size eect for simulations corresponding to the sealing of fractures of a quartz-rich rock at 5 km depth. The initial fracture porosity after an earthquake is the same in all cases. The rate of porosity decrease changes as the mean distance between the microcracks (Lsz) varies from 10 mm (a lot of small cracks), 20, 50, 100, 200, 500, to 1000 mm (few larger cracks). The aperture of microcracks (Lv) varies from 0.6, 1.2, 3, 6, 12, 30 to 60 mm, respectively. The rate of crack sealing is much higher for a large number of small cracks than fewer larger ones. (b) Size eect for the sealing of fractures of a calcite-rich rock at 4 km depth. In this case also, the rate of sealing decreases when the mean distance between cracks increases (from 50, 100, 200, 500, to 1000 mm) and the crack thickness increases (3, 6, 12, 30, 60 mm, respectively).
3.3. Sensitivity to various parameters tance between stylolites and cracks, the slower is the transport from the dissolution site to the precipitation In these simulations, the least constrained par- surface. Eect of the spacing between veins is given in ameters are the coecient of diusion along grain con- Fig. 7. From 10 mm to 1 mm, the strain rate decreases tacts or along the stylolites, the grain size that can by six orders of magnitude. This is clearly the major vary locally in the fault; the crack width, and the eect and careful analysis of the main distance of mass mean distance between the veins (which imposes the transfer in natural deformation is required. area of dissolution under stress and thus the mean dis- For distances comparable to those in Fig. 1 (100± tance of diusion along a trapped ¯uid). When these 200 mm between stylolites), the sealing time of the frac- parameters vary, the time for fracture porosity closure ture network in quartz-rich rocks at 4±6 km varies is modi®ed. between 500 and 100 000 y depending on the value of Decreasing the coecient of diusion of solutes the coecient of diusion. The lower value is close to from stylolites to open veins at a constant depth slows the recurrence time of major earthquakes along active down pressure solution (Fig. 6). If the diusion is as faults (Scholz, 1990). fast as in bulk water, the rate of pressure solution is The contributions of the dierent processes of press- two orders of magnitude higher. This parameter is, at ure in the upper crust are summarized in Fig. 7 for the this writing, the least constraint one in pressure sol- case of pure quartz or pure carbonate rocks. The rate ution models. It should strongly depend on the ¯atness of pressure solution in quartz rocks increases with of grain interfaces. We expect that new experiments in depth both in the gouge and in the fractures with the the coming years will help de®ne more accurate values dierence that the gouge compaction is much faster of this parameter and its dependence with temperature, than the fracture sealing because of the dierence in and perhaps stress. the transport distance and the thickness of the dis- solved and precipitated layers. 3.4. Eects of grain size and vein spacing For carbonate rocks, the dependence of pressure sol- ution with depth is more complex. The strain rate of There is also a size eect, depending on the mean pressure solution increases with depth, from 1 to 5 km. distance between veins (Fig. 7). A dense array of thin This rates decreases for greater depths because of a cracks will close much faster than a loose array of decrease of calcite solubility with temperature. The thicker cracks (Gratier et al., 1999). The larger the dis- strain rate curves of carbonates rocks cross the curve F. Renard et al. / Journal of Structural Geology 22 (2000) 1395±1407 1405 of quartz rocks at a temperature of 220±2308C, corre- 0 sponding to depths of 7 km for a gradient of tempera- gouge, carbonate ture of 308/km (Fig. 8). Such calculations are in 1 agreement with geological observations showing that calcite is mobile at shallow depths whereas quartz is 2 more mobile at greater depths. The transition between 3 these two ®elds is mainly controlled by temperature gouge, quartz and by the chemistry of rocks and ¯uids. 4 crack, quartz
5 crack, carbonate 4. Discussion 6 Depth (km)
Evans and Chester (1995) have observed that ¯uid- 7 driven reactions and veining occurred during faulting, but are localized in a small region near the fault plane. 8 For example, in the San Gabriel fault, geochemical data indicate that the system could have been opened 9 to ¯uids for some time after an earthquake. As a ®ne- grained gouge compacts fast (several years according 10-14 10-13 10-12 10-11 10-10 10-9 10-8 to our calculations), ¯uid could not circulate eciently Deformation rate (s-1) during a long period there, compared to open veins for which the kinetics of closing are much slower (sev- Fig. 8. Summary of the dierent processes of ductile deformation in eral hundred to several million years). The distance to an upper crust with carbonate rocks and quartz-rich rocks between 2 and 9 km. The two mechanisms of deformation presented here are the fault over which the ¯uid can circulate should cor- compaction of a ®ne-grained gouge and sealing of open veins respond to the area damaged by microfracturing coupled with dissolution along stylolites. Gouge compaction is much around a fault during a seismic event. faster than crack sealing and there is a clear control of the rate of In addition, geodetic observations of post-seismic deformation by the lithology. Such simulations are performed with deformation in the Northridge area show the presence textural and chemical variables that can be measured independently. Even if there can be a large uncertainty on some of the input par- of a creep deformation in the upper crust, associated ameters (rock texture, kinetics and thermodynamics data), it seems with displacements on the surface over periods of sev- that the time scale of pressure solution creep can be relatively small eral years (Donnellan and Lyzenga, 1998). These ob- and should play a role in the seismic cycle. Such calculations indicate servations are in agreement with a deformation also that there exists a transition depth in the crust. One passes from controlled by ¯uids and mechano-chemical processes carbonate deformation at shallow depths to deformation accommo- dated by quartz-rich rocks at greater depths; the strain rates curves, in the upper crust. Our modeling indicates that such crossing each other at a depth controlled by temperature, around creep deformation can be caused by the two main pro- 2208C in these simulations. cesses of pressure solution: intergranular pressure sol- ution, and dissolution along stylolite coupled to vein sealing. fast enough to induce a zone of low permeability pro- Our calculations also suggest that permeability re- moting high pore pressure between 2 and 5 km depth. duction around active faults in California should occur Extrapolating this model to natural systems is rapidly between 0 and 4 km because of calcite precipi- clearly limited by the crucial eect of the geometry of tation in open fractures. Below 7±8 km, the kinetics of the diusion pathways. Geometric parameters such as quartz dissolution and precipitation are fast enough to the spacing between the cracks are not easy to estimate allow quick vein sealing. As a consequence, there from natural observations. This is probably the main should exist a region in the crust characterized by source of uncertainties in this approach. Careful stu- slower porosity and permeability reduction, between 4 dies must be performed in order to improve our and 7 km. At this depth interval the decrease of calcite knowledge of the three-dimensional geometry of such solubility slows down the process of vein sealing and diusion pathways. the quartz kinetics are too slow to close fractures e- Another uncertainty comes from the poor constraint ciently. Such calculations indicate that the local lithol- on the relationship between stress and crack-seal kin- ogy is a crucial factor in controlling the rate of etics. This needs a major experimental eort. pressure solution-induced deformation in the crust. The solubility crossover has signi®cant implications for the distribution of pore pressure and its evolution 5. Conclusions during the seismic cycle. For example if calcite is mobilized at shallow depth, localized crack sealing is Such calculations stress that viscous compaction of 1406 F. Renard et al. / Journal of Structural Geology 22 (2000) 1395±1407 the rocks inside a fault plays an important role during slip and upper crustal deformation following the Northridge the seismic cycle and show that brittle and ductile de- earthquake. Journal of Geophysical Research 103, 21285±21297. formation can interact in the upper crust (Gratier et Dove, P.M., 1994. The dissolution kinetics of quartz in sodium chloride solutions at 258C to 3008C. American Journal of Science al., 1999). Two time-scales emerge from our calcu- 294, 665±712. lations based on natural observations: pervasive press- Evans, J.P., Chester, F.M., 1995. Fluid±rock interaction in faults of ure solution at the grain scale in the gouge is much the San Andreas system: Inferences form San Gabriel fault rock faster than pressure solution along stylolites and as- geochemistry and microstructures. Journal of Geophysical sociated precipitation in veins. Following an earth- Research 100, 13007±13020. Farver, J.R., Yund, R.Y., 1995. Interphase boundary diusion of quake, the permeability loss in the gouge, due to oxygen and potassium in K-feldspar/quartz aggregates. intergranular pressure solution, should be much faster Geochimica and Cosmochimica Acta 59, 3697±3705. than permeability variations due to fracture sealing. Farver, J.R., Yund, R.Y., 1999. Oxygen bulk diusion measurements This suggests that open veins in the country rocks and TEM characterization of a natural ultramylonite for ¯uid around the fault plane mainly control ¯uid circulation. transport in mica-bearing rocks. Journal of Metamorphic Fluids play a key role in the viscous relaxation Geology 17, 669±684. Fisher, D.M., Brantley, S.L., Everett, M., Dzvonik, J., 1995. Cyclic occurring in the upper crust after an earthquake. ¯uid ¯ow through a regionally extensive fracture network within In the Northridge area, the post-seismic slip the Kodiak accretionary prism. Journal of Geophysical Research observed after the 1994 earthquake can reach several 100, 12881±12894. centimeters per year and is a witness of a `soft upper Fisher, D.M., Brantley, S.L., 1992. Models of quartz overgrowth crust'. Common explanations of this relaxation involve and vein formation; deformation and episodic ¯uid ¯ow in an the ¯ow of lower crustal material (Donnellan and ancient subduction zone. Journal of Geophysical Research 97, 20043±20061. Lyzenga, 1998). We suggest here that pressure solution Fournier, R.O., 1996. Compressive and tensile failures at high ¯uid may also be a mechanism of stress relaxation in the pressure where preexisting fractures have cohesive strength, with gouge and in the country rock surrounding the fault application to the San Andreas Fault. Journal of Geophysical over periods of several days to hundreds of years after Research 101, 25499±25509. a major earthquake. We stress also that more exper- Gibbs, J.W. 1877. On the equilibrium of heterogeneous substances. Transactions of the Connecticut Academy 3, 108-248 and 343- iments on pressure solution, and particularly on the 524. rate of diusion along grain boundaries, should help Gratier, J.P., Guiguet, R., 1986. Experimental pressure solution on reducing the uncertainty concerning the time scales of quartz grains: the crucial eect of the nature of the ¯uid. Journal this creep deformation. of Structural Geology 8, 845±856. Gratier, J.P., 1993. Experimental pressure solution of halite by an indenter technique. Geophysical Research Letters 20, 1647±1650. Gratier, J.P., Gamond, J.F., 1990. Transition between seismic and Acknowledgements aseismic deformation in the upper crust. In: Knipe, R.J., Rutter, E.H. (Eds.), Deformation, Mechanisms, Rheology and Tectonics, This project has been supported by the Norwegian Geological Society Special Publication, 54, pp. 461±473. Research Council through grant number 113354/420 Gratier, J.P., Chen, T., Hellmann, R., 1994. Pressure solution as a to the Fluid±Rock Interactions. We would like to mechanism for crack sealing around faults, Proceedings of Workshop LXIII on the Mechanical involvement of ¯uids in thank B. Parry and J. Hadizedeh for helpful comments faulting, 94-228. USGS open-®le report, Menlo Park, CA, pp. and J.P. Evans for his editorial help. 279±300. Gratier, J.P., Renard, F., Labaume, P., 1999. How pressure solution creep and fracturing process interact in the upper crust to make it References behave in both a viscous and brittle manner. Journal of Structural Geology 21, 1189±1197. Gudmundsson, A., 1999. Fluid overpressure and stress drop in fault Applin, K.R., 1987. The diusion of dissolved silica in dilute aqu- zones. Geophysical Research Letters 26, 115±118. eous solution. Geochimica and Cosmochimica Acta 51, 2147± Henderson, J.R., Maillot, B., 1997. The in¯uence of ¯uid ¯ow in 2151. fault zones on patterns of seismicity: a numerical investigation. Brace, W.F., Byerlee, J.D., 1966. Stick-slip as a mechanism for earth- Journal of Geophysical Research 102, 2915±2924. quakes. Science 153, 990±992. Byerlee, J.D., 1990. Friction, overpressure and fault normal com- Hickman, S.H., Evans, B., 1991. Experimental pressure solution in pression. Geophysical Research Letters 17, 2109±2112. haliteÐThe eect of grain interphase boundary structure. Journal Chester, F.M., 1995. A rheologic model for wet crust applied to of the Geological Society of London 148, 549±560. strike-slip faults. Journal of Geophysical Research 100, 13033± Hickman, S., Sibson, R., Bruhn, R., 1995. Introduction to the special 13044. issue: mechanical involvement of ¯uid in faulting. Journal of Dewers, T., Ortoleva, P., 1990. A coupled reaction/transport/mech- Geophysical Research 100, 12831±12840. anical model for intergranular pressure solution stylolites, and Janssen, C., Michel, W., Bau, M., Luders, V., Muhle, K., 1997. The dierential compaction and cementation in clean sandstones. North Anatolian Fault Zone and the role of ¯uids in seismogenic Geochimica and Cosmochimica Acta 54, 1609±1625. deformation. Journal of Geology 105, 387±403. Donnellan, A., Lyzenga, D.A., 1998. GPS observations of fault after Johnson, J.W., Oelkers, E.H., Helgeson, H.C., 1992. SUPCRT92ÐA software package for calculating the standard molal thermodyn- amic properties of minerals, gases, aqueous species, and reactions F. Renard et al. / Journal of Structural Geology 22 (2000) 1395±1407 1407
from 1 bar to 500 bar and 08C to 10008C. Computers and sandstones: In¯uence of clays and dependence on temperature Geosciences 18, 899-947. and stress. Tectonophysics 280, 257±266. Lachenbruch, A.H., 1980. Frictional heating, ¯uid pressure and the Renard, F., Park, A., Ortoleva, P., Gratier, J.P., 1999. An integrated resistance to fault motion. Journal of Geophysical Research 85, model for transitional pressure solution in sandstones. 6097±6112. Tectonophysics 312, 97±115. Lockner, D.A., Byerlee, J.D., 1995. An earthquake instability model Rimstidt, J.D., Barnes, H.L., 1980. The kinetics of silica±water reac- based on faults containing high ¯uid-pressure compartments. tions. Geochimica and Cosmochimica Acta 44, 1683±1699. Pure and Applied Geophysics 145, 717±745. Rice, J.R., 1975. Continuum mechanics and thermodynamics of plas- Matthai, S.K., Fisher, G., 1996. Quantitative modeling of fault-¯uid- ticity in relation to microscale deformation mechanisms. In: discharge and fault-dilation-induced ¯uid-pressure variations in Argon, A.S. (Ed.), Constitutive Equations in Plasticity. MIT the seismogenic zone. Geology 24, 183±186. Press, Cambridge, Mass, pp. 23±79. Marone, C., Raleigh, C.B., Scholz, C.H., 1990. Frictional behavior Rice, J.R., 1992. Fault stress states, pore pressure distributions, and and constitutive modeling of simulated fault gouge. Journal of the weakness of the San Andreas Fault. In: Evans, B., Wong, T.- Geophysical Research 95, 8441±8452. F. (Eds.), Fault Mechanics and Transport Properties in Rocks. Miller, S.A., Nur, A., Olgaard, D.L., 1996. Earthquakes as a coupled Academic Press, London, pp. 475±503. stress±high pore pressure dynamical system. Geophysical Rudnicki, J.W., Chen, C.H., 1988. Stabilization of rapid frictional Research Letters 23, 197±200. slip on a weakening fault by dilatant hardening. Journal of Oelkers, E.H., Bjùrkum, P.A., Murphy, W.M., 1996. A petrographic Geophysical Research 93, 4745±4757. and computational investigation of quartz cementation and por- Rutter, E., 1976. The kinetics of rock deformation by pressure sol- osity reduction in North Sea sandstones. American Journal of ution. Philisophical Transactions of the Royal Society of London Science 296, 420±452. 283, 203±219. Paterson, M.S., 1973. Nonhydrostatic thermodynamics and its geolo- Segall, P., Rice, J.R., 1995. Dilatancy, compaction, and slip instabil- gic applications. Reviews of Geophysics and Space Physics 11, ity of a ¯uid-in®ltrated fault. Journal of Geophysical Research 355±389. 100, 22155±22171. Pili, E., et al. 1998. Isotope constraints on the involvement of ¯uids Scholz, C.H., 1990. The Mechanics of Earthquakes and Faulting. in the San Andreas Fault. Abstracts of Proceedings, AGU 1998 Cambridge University Press, Cambridge. Spring Meeting. Shimizu, I., 1995. Kinetics of pressure solution creep in quartz: Plummer, L.N., Wigley, T.L.M., Parkhurst, D.L., 1978. The kinetics theoretical considerations. Tectonophysics 245, 121±134.
of calcite dissolution in CO2±water systems at 58 to 608C and 0.0 Sleep, N.H., Blanpied, M.L., 1994. Ductile creep and compaction: A to 1.0 atm CO2. American Journal of Science 278, 179±216. mechanism for transiently increasing ¯uid pressure in mostly Plummer, L.N., Parkhurst, D.L, Wigley, T.L.M., 1979. Critical sealed fault zones. Pure and Applied Geophysics 143, 9±40. review of the kinetics of calcite dissolution and precipitation. In: Spiers, C.J., Schutjens, P.M., Brzesowsky, R.H., Peach, C.J., Jenne, E.A. (Ed.), Chemical Modeling in aqueous systems, Liezenberg, J.L., Zwart, H.J., 1990. Experimental determination American Chemical Symposium Society Series, 93, pp. 537±573. of constitutive parameters governing creep of rocksalt by pressure Plummer, L.N., Busenberg, E., 1982. The solubilities of calcite, ara- solution. In: Knipe, R.J., Rutter, E.H. (Eds.), Deformation
gonite, and vaterite in CO2±H2O solutions between 0 and 908C, Mechanisms, Rheology and Tectonics, Geological Society Special and an evaluation of the aqueous model for the system CaCO3± Publication, 54, pp. 215±227. CO2±H2O. Geochimica et Cosmochimica Acta 46, 1011±1040. Walderhaug, O., 1994. Precipitation rates for quartz cement in sand- Raj, R., Chyung, C.K., 1981. Solution-precipitation creep in glass stones determined by ¯uid-inclusion microthermometry and tem- ceramics. Acta Metallurgica 29, 159±166. perature±history modeling. Journal of Sedimentary Research 64, Raj, R., 1982. Creep in polycristalline aggregates by matter transport 324±333. through a liquid phase. Journal of Geophysical Research 87, Weyl, P.K., 1959. Pressure solution and the force of crystallization: 4731±4739. A phenomenological theory. Journal of Geophysical Research 69, Renard, F., Ortoleva, P., 1997. Water ®lms at grain±grain contacts: 2001±2025. Debye±Huckel, osmotic model of stress, salinity and mineralogy Yamashita, T., 1997. Mechanical eect of ¯uid migration on the dependence. Geochimica and Cosmochimica Acta 61, 1963±1970. complexity of seismicity. Journal Geophysical Research 102, Renard, F., Ortoleva, P., Gratier, J.P., 1997. Pressure solution in 17797±17806.