Modeling the growth of stylolites in sedimentary rocks

HAL Id: hal-00701974

https://hal.archives-ouvertes.fr/hal-00701974

Preprint submitted on 28 May 2012

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Alexandra Rolland, Renaud Toussaint, Patrick Baud, Jean Schmittbuhl, Nathalie Conil, Daniel Koehn, Francois Renard, Jean-Pierre Gratier

To cite this version:

Alexandra Rolland, Renaud Toussaint, Patrick Baud, Jean Schmittbuhl, Nathalie Conil, et al.. Mod- eling the growth of stylolites in sedimentary rocks. 2012. �hal-00701974�

Modeling the growth of stylolites in sedimentary

1

rocks.

2

Alexandra Rolland

, Renaud Toussaint

, Patrick Baud

, Jean Schmittbuhl

, Nathalie Conil

, Daniel Koehn

, Fran¸cois Renard

, Jean-Pierre Gratier

Alexandra Rolland, Institut de Physique du Globe, Universit´e de Strasbourg, 5 rue Ren´e Descartes, 67084 cedex, Strasbourg, France (alexandra.rolland@unistra.fr)

Renaud Toussaint, Institut de Physique du Globe, Universit´e de Strasbourg, 5 rue Ren´e Descartes, 67084 cedex, Strasbourg, France (renaud.toussaint@unistra.fr)

Patrick Baud, Institut de Physique du Globe, Universit´e de Strasbourg, 5 rue Ren´e Descartes, 67084 cedex, Strasbourg, France (patrick.baud@unistra.fr)

Jean Schmittbuhl, Institut de Physique du Globe, Universit´e de Strasbourg, 5 rue Ren´e Descartes, 67084 cedex, Strasbourg, France (jean.schmittbuhl@unistra.fr)

Nathalie Conil, Agence Nationale pour la gestion des D´echets Radioactifs, Centre de Meuse/Haute-Marne, Route D´epartementale 960, BP 9, 55290, Bure, France

Daniel Koehn, University of Glasgow, Scotland

Fran¸cois Renard, ISTerre, Universit´e Joseph Fourier, Grenoble, France Jean-Pierre Gratier, ISTerre, Universit´e Joseph Fourier, Grenoble, France

1

Institut de Physique du Globe,

Abstract. Stylolites are ubiquitous pressure-solution seams found in sed-

3

imentary rocks. Their morphology is shown to follow two self-affine regimes:

4

analyzing the scaling properties of their height over their average direction

5

shows that at small scale, they are self-affine surfaces with a Hurst exponent

6

around 1, and at large scale, they follow another self-affine scaling with Hurst

7

exponent around 0.5. In the present paper we show theoretically the influ-

8

ence of the main principal stress and the local geometry of the stylolitic in-

9

terface on the dissolution reaction rate. We compute how it is affected by

10

the deviation between the principal stress axis, and the local interface be-

11

tween the rock and the soft material in the stylolite. The free energy enter-

12

ing in the dissolution reaction kinetics is expressed from the surface energy

13

term, and via integration from the stress perturbations due to these local mis-

14

alignments.The resulting model shows the interface evolution at different stress

15

conditions. In the stylolitic case, i.e. when the main principal stress is nor-

16

Universit´e de Strasbourg, France

2

Andra, Bure, France

3

School of Geographical and Earth Sciences, University of Glasgow, France

4

ISTerre, University Joseph Fourier - Grenoble I and CNRS, France

5

PGP, University of Oslo, Norway

mal to the interface, two different stabilizing terms dominate at small and

17

large scales which are linked respectively to the surface energy and to the

18

elastic interactions. Integrating the presence of small scale heterogeneities

19

related to the rock properties of the grains in the model leads to the formu-

20

lation of a Langevin equation predicting the dynamic evolution of the sur-

21

face. This equation leads to saturated surfaces obeying the two observed scal-

22

ing laws. Analytical and numerical analysis of this surface evolution model

23

shows that the cross-over length separating both scaling regimes depends di-

24

rectly on the applied far-field stress magnitude. This method gives the ba-

25

sis for the development of a paleostress magnitude marker. We apply the com-

26

putation of this marker, i.e. the morphological analysis, on a stylolite found

27

in the Dogger limestone layer located in the neighborhood of the Andra Un-

28

derground Research Laboratory at Bure (Eastern France). The results are

29

consistent with the two scaling regimes expected, and the practical deter-

30

mination of the major principal paleostress, from the estimation of a cross-

31

over length, is illustrated on this example.

32

1. Introduction

Stylolites are undulated surfaces resulting from localized stress-driven dissolution of

33

some minerals of the rock. Insoluble minerals as clay particles, oxides and organic mat-

34

ters are concentrated in the interface and make stylolites visible. Bathurst [1987] describes

35

stylolites as serrated interfaces with an amplitude greater than the diameter of the tran-

36

sected grains giving them a sutured appearance. He makes a difference with dissolution

37

seams or ’flaser’ which are smooth, undulating, lacking in sutures and fitting around grains

38

instead of cutting through them. Stylolites are most often found in carbonates [Stock-

39

dale, 1922, 1926, 1936, 1943; Dunnington , 1954; Bushinskiy , 1961; Park and Schot , 1968;

40

Bathurst, 1971; Buxton and Sibley , 1981; Railsback , 1993] but also in sandstones [Young,

41

1945; Heald , 1955], shales [Wright and Platt, 1982; Rutter , 1983], cherts [Bushinskiy ,

42

1961; Iijima, 1979; Cox and Whitford-Stark , 1987] and sometimes in coal [Stutzer , 1940].

43

Stylolites are divided in two groups according (i) to their orientation with respect to the

44

bedding of the surrounding rock or (ii) to the orientation of their ’tooth’ with respect to

45

the mean plane of the stylolite. The first group shows two types of orientation: stylolites

46

parallel to the bedding plane, designated as sedimentary, and formed under the lithostatic

47

pressure and stylolites oblique or even perpendicular to the bedding, designated as tec-

48

tonic, and depending on the maximum tectonic stress. The tooth orientation is in both

49

cases an indicator of the direction of the incremental displacement which is parallel to

50

the major principal stress in co-axial deformation. The stylolites of the second group are

51

called ’slickolites’ [Ebner et al., 2010a]. They develop when there is a preferential plane

52

for their growth (bedding or fracture). In this case, the stress is not perpendicular to the

53

mean plane of the stylolite [Stockdale, 1922], but the edges of the tooth are subparallel to

54

the maximum principal stress axis. Various studies [Park and Schot, 1968; Renard et al.,

55

1997, 2001; Andr´e, 2003; Aharonov and Katsman, 2009] suggest that many parameters

56

play an important role in the stylolite growth such as confining pressure, deviatoric stress,

57

fluid pressure, temperature, shape and assemblage of grains, anisotropy of minerals, rates

58

of dissolution and presence of clay (acting potentially as catalyst for the dissolution).

59

Only few papers report experiments about stylolites development. Indeed, they are

60

inherently difficult to reproduce as the kinetics of pressure-solution processes is very slow

61

[Rutter , 1976]. Experiments were conducted either on aggregates [Cox and Paterson,

62

1991; Den Brok and Morel , 2001; Renard et al., 2001; Gratier et al., 2005] or with in-

63

denter techniques [Gratier and Guiguet , 1986; Gratier , 1993; Gratier et al., 2004; Dysthe

64

et al., 2002, 2003; Karcz et al., 2008]. Dysthe et al. [2002, 2003] used an indenter technique

65

where a sodium chloride crystal was kept in contact with a piston at given pressure and

66

temperature for several months. A fluid at compositional equilibrium with the crystal

67

is trapped between the sample and the indenter. The contact evolved due to pressure-

68

solution during the indentation. A power law time dependence with an exponent value of

69

1/3 as in Andrade creep law was shown to control the indentation rate. The observed mi-

70

crostructures in the contact seem to be different from stylolites. Karcz et al. [2008] loaded

71

a halite cone-shaped indenter against a flat silicate surface immersed in an undersatu-

72

rated brine. Using confocal microscopy techniques, they observed that the evolution of

73

the system is dictated by an interaction between two deformation mechanisms: undercut-

74

ting dissolution reducing the area of the contact and plastic flow increasing it. Recently,

75

similar experiments were carried out with a brine at chemical equilibrium with the crystal

76

[Laronne Ben-Itzhak , 2011]. Emerging evolving islands and channels were observed at

77

the contact. Such islands and channels structures were previously observed at the con-

78

tacts during experiments on aggregates [Schutjens and Spiers , 1999; Den Brok and Spiers,

79

1991]. Other experiments on aggregates were performed by Gratier et al. [2005]. They

80

loaded layers of fine quartz sand grains. The experiments lasted several months at 350

C,

81

under 50 MPa of differential stress and in presence of an aqueous silica solution. Microsty-

82

lolites were created for the first time in the laboratory at the stressed contacts between the

83

quartz grains. An interesting observation is that the stylolites peaks are always located

84

in front of dislocation pits. Consequently, stylolites appear to be localized by the hetero-

85

geneities of the mineral. Den Brok and Morel [2001] loaded elastically K-alum crystals at

86

a controled temperature and in a saturated K-alum solution. A hole was drilled in the

87

middle of the crystals to provide an elastic strain gradient. They observed macroscopic

88

etch grooves on the originally smooth free surfaces of the soluble crystals which disappear

89

when removing the stress. Koehn et al. [2004] stressed crystals of NaClO

in a NaClO

90

solution at room temperature. Parallel dissolution grooves developed on their free surface

91

in a 1D geometry to a 2D geometry with the coarsening of the pattern. The pressure-

92

solution process slowed down or stopped progressively with the increasing concentration

93

of the solution during the experiments. Gratier et al. [2004] used a similar technique in

94

which a sample of Bure claystone was kept in contact with a piston, with a saturated

95

brine in the contact, at an imposed pressure and temperature for several months. No

96

evidence of localized pressure-solution (dissolution seam) was observed in this case, grain

97

to grain sliding being more efficient in presence of clay. Renard et al. [2001], studied

98

chemical compaction of aggregates of halite (salt) mixed with clay. They showed that

99

clay particles enhance pressure-solution. Moreover, Renard et al. [1997] studied the effect

100

of clay on clay-rich sandstones. They suggested that pressure-solution is enhanced by clay

101

because a thick film of water is preserved between clay particles. They also concluded

102

that the depth determines the limiting factor for the process: at great depth, the water

103

film between grains should be thinner and diffusion limits the process. Conversely, at low

104

depth water films are bigger, transport is easier and the reaction kinetics is the limiting

105

factor.

106

The clay particles effect on pressure-solution was recently simulated in numerical mod-

107

eling. Aharonov and Katsman [2009] used the two-dimensional Spring Network Model to

108

study the stylolites growth in a medium with a uniform clay distribution. They showed

109

that clay plays a role of enhancing pressure-solution and that stylolites propagation is

110

possible only when both pressure-solution and clay-enhanced dissolution operate together.

111

Koehn et al. [2007] developed a new discrete simulation technique that reproduces suc-

112

cessfully the roughening of stylolites from a preferential existing surface with no clay. This

113

model is based on molecular dynamics, with a dissolution speed depending on the local

114

free energy that includes stress dependent terms and surface energy terms. Two different

115

spatial regimes arise from this modeling: a small-scale regime where surface energy is

116

dominant with significant fluctuations of the roughness and a large-scale regime where

117

elastic energy dominates. The dependence on the cross-over scale between both regime

118

on the imposed stress has been recently investigated numerically [Koehn, 2012]. This

119

model shows that the growth of the stylolite tooth follows the main compressive stress

120

direction. The nature and structure of the small scale disorder for the dissolution prop-

121

erties of grains were systematically analyzed [Ebner et al., 2009a]. Moreover, Ebner et al.

122

[2010b] performed detailed microstructural analysis to investigate the interplay between

123

this disorder and the compositional nature of the grains surrounding a stylolite.

124

Stylolites are localised features for which deformation is purely compactant as for com-

125

paction bands [Mollema and Antonellini, 1996; Baud et al., 2004; Katsman et al., 2006b;

126

Tembe et al., 2008]. Stylolites and compaction bands development was modeled as ant-

127

icracks or anti-mode I fracture [Fletcher and Pollard , 1981; Rispoli, 1981; Mollema and

128

Antonellini , 1996]. Fletcher and Pollard [1981] assume that the rate of pressure-solution

129

is only a function of the normal stress. They observed an elliptic dissolution pattern i.e.

130

more dissolution in the central part of stylolites than at the tips. With these observations

131

they proposed an analogy between propagation of stylolites and propagation of mode I

132

fractures. They observed that the relative displacement between the sides of a stylolite

133

should have the opposite sign than that of a crack, and thus termed their model an an-

134

ticrack. Note however that cracks can bear zero surface traction, contrary to stylolites.

135

This distinction between crack solutions and stylolites was introduced, and it was shown

136

by Katsman et al. [2006a] that, as compaction bands, stylolites are Localized Volume

137

Reduction zones (LVR). The shape of the displacement along stylolites, and how the

138

stress perturbation can be determined from the concept of LVR, is discussed in details by

139

Katsman [2010].

140

In LVR where the dissolution amount is constant across the surface of the LVR, as for a

141

compaction band, the stress enhancement was shown to be that of a dislocation [Katsman

142

et al., 2006a]. In later models Katsman [2010], it was shown that if more dissolution is

143

allowed in the center of a stylolite, another type of stress enhancement, with a dependence

144

on the distance to the tips analogous to the one for a crack (rather than to a dislocation),

145

can be observed it is given by the Eshelby inclusion problem. Such an increase of the

146

dissolution in the center of a stylolite, where the dissolution does not stop in the already

147

dissolved zone in the middle of the stylolite, can be observed in models with a positive

148

feedback do the dissolution, as for example the one that can be modeled from a clay

149

concentration mechanism [Aharonov and Katsman, 2009].

150

In general, in stylolites, the stress concentrates at the tips and the largest stress is

151

perpendicular to the stylolites. Recent models [Koehn et al., 2007; Ebner et al., 2009b;

152

Zhou and Aydin, 2010] suggest that a higher stress concentration at the top of the tooth

153

should be responsible of localized high rates of dissolution. Benedicto and Schultz [2010]

154

investigated the topography of stylolites (along-strike trace length, maximum and average

155

amplitudes) from the damaged zone of the Gubbio normal fault zone in central Italy. They

156

showed that the amount of contractional strain accommodated by stylolites as well as their

157

length and their number increase according to the topography parameters. Analyses of

158

cores from boreholes reveal also an increase in stylolite abundance with depth [Lind ,

159

1993]. Fabricius and Borre [2007] compared formations of chalk from boreholes on the

160

Ontong Java Plateau and in the central North Sea. They showed that the burial stress

161

and the temperature play distinct roles in the burial diagenesis and porosity development

162

of chalk. Pressure-solution and physical compaction are controled by the burial stress

163

while the temperature controls recrystallization and cementation. Moreover, Lind [1993]

164

suggests that mineralogical anomaly is an initializing factor in stylolite formation such as

165

burrows, shale clasts or flaser structures. Many studies were conducted on the morphology

166

of sedimentary stylolites [Renard et al., 2004; Brouste et al., 2007; Ebner et al., 2009b].

167

Morphology analyses can be done on 1D profiles or 2D opened surfaces. They consist

168

on studying a stylolitic profile or surface height variations (standard deviation, height

169

differences, power spectrum, average wavelet coefficient spectrum, etc.) over different

170

scales [Schmittbuhl et al., 1995, 2004; Renard et al., 2004]. These analyses reveal two

171

distinct scaling regimes that could be described by power laws. The power laws are

172

function of a roughness exponent also called Hurst exponent inferred to be 1 and 0.5 for

173

small and large scale respectively [Renard et al., 2004; Schmittbuhl et al., 2004; Brouste

174

et al., 2007; Ebner et al., 2009b]. The two regimes are separated by a cross-over length

175

typically around 1 mm [Renard et al., 2004; Schmittbuhl et al., 2004]. For sedimentary

176

stylolites, the two dimensional (2D) analysis of their surface does not show any significant

177

inplane anisotropy reflecting the fact that horizontal stresses are isotropic. Ebner et al.

178

[2010a] observed that the profiles of tectonic stylolites show the same geometric attributes

179

as sedimentary ones. Two different regimes are also observed with Hurst exponent around

180

1 and 0.5 for small and large scale respectively. However, for tectonic stylolites, the 2D

181

analysis revealed an anisotropy of the cross-over length which varies with the direction in

182

the plane of stylolites. Ebner et al. [2010a] argue that this anisotropy develops because

183

the stylolite roughens in an anisotropic inplane stress field. The vertical and inplane

184

horizontal stresses are significantly differents. In recent papers, stylolites are presented as

185

fossilized signatures of the stress field [Renard et al., 2004; Schmittbuhl et al., 2004; Ebner

186

et al., 2009b, 2010a]. The existence of two scaling regimes for sedimentary stylolites was

187

shown in Schmittbuhl et al. [2004] where a brief theoretical derivation was performed.

188

It was shown that the cross-over length between both scaling regimes is expected to

189

be dependent on the stress acting on the stylolite during its growth. Their conclusion

190

was that stylolite morphology can be used as a paleostress magnitude indicator. This

191

conclusion was later probed independently on two types of approaches: first, on field data

192

sampled from the same formation at different heights, Ebner et al. [2009b] showed that the

193

measured cross-over length in the morphology followed the expected scaling with the burial

194

stress, evaluated from the position in the formation. Next, discrete numerical simulations

195

were carried out at different stress magnitudes, allowing for the dissolution of grains along

196

the fluid/rock interface, with free energy depending on interfacial tension and local stress.

197

It was shown that the two expected scaling regimes were observed [Koehn et al., 2007;

198

Koehn, 2012], and that the cross-over length followed the predicted dependence on the

199

far-field stress amplitude [Koehn, 2012].

200

Interfaces between solids and fluids are related to models of stylolitization. In the case

201

where a solid in contact with a fluid is stressed, an instability due to pressure-solution

202

was shown theoretically to exist and is called the Asaro-Tiller-Grinfeld (ATG) instability

203

[Renard et al., 2004]. In models of dissolving surfaces with a stress imposed to a solid in

204

contact with a fluid at chemical equilibrium, this instability leads to the growth of initial

205

large scale modulations of the surface with a wavelength selection obtained through a

206

fastest growing mode. The basic equation depends on the particular boundary conditions

207

e.g. when two solids with different elastic properties are in contact and submitted to a

208

stress, the interface can undergo a fingering instability led by the contrast between the

209

free energies applied to both solids [Angheluta et al., 2008, 2009, 2010]. The stability

210

analysis can be performed theoretically from expressions for the kinetics using local free

211

energy criteria for the reaction rate [Renard et al., 2004; Schmittbuhl et al., 2004], or

212

global ones [Bonnetier et al., 2009; Angheluta et al., 2008]. Depending on the boundary

213

conditions, this situation is also found to be unstable for perturbations exceeding a certain

214

wavelength, leading to fingering (as e.g. with large stress tangential to a fluid interface,

215

or a stress normal to fluid interfaces and lateral periodic boundary conditions [Bonnetier

216

et al., 2009]). With other boundary conditions, the surface energy and elastic interactions

217

are found to stabilize the interfaces, which are only destabilized by material noise due to

218

heterogeneities [Schmittbuhl et al., 2004; Koehn et al., 2007]. We will argue in details in the

219

discussion section about the different possibilities applied to the geometry of stylolites,

220

and the fact that stylolites displaying self-affine scaling laws for their height at large

221

scale are compatible with the stabilizing character of elastic forces at large scale. This

222

manuscript provides the technical development and details that lead to the final result

223

that was previously published without derivation, in a condensed form [Schmittbuhl et al.,

224

2004]. It also compares the result of the analytical development to a direct numerical

225

simulation.

226

In this paper we concentrate on the following questions: (i) Is the elastic energy sta-

227

bilizing or destabilizing? (ii) What is the significance of the obtained paleostress values?

228

To answer to these questions, (i) we derive the details of the computation leading to the

229

link between the paleostress magnitude and the cross-over length between the two scaling

230

regimes. This is performed by a perturbative analysis of the elastic energy around an

231

interface slightly wavy and unaligned with one of the principal stresses. Then we show in

232

details that the mechanics and chemistry allow to relate the small and large scale behavior

233

of stylolites to known models, with Hurst exponents corresponding to the observed ones.

234

(ii) We finally present and discuss an application in relation with the geological context.

235

This is made on a stylolite from the Bure carbonates and it shows how the predicted

236

scaling regimes can be found, and how to determinate the paleostress from the extracted

237

cross-over length.

238

2. Analytical approach : Continuous elastostatic model for stylolite propagation

The rough morphology of stylolites arises from the disorder present in a rock and its

239

impact on the pressure-solution process. This disorder is spatially linked to the grains con-

240

stituting the rock. To understand the impact of this disorder on the chemico-mechanical

241

coupling, we will consider the following simplified geometry: the initial stage of the sty-

242

lolite is modeled as an elongated fluid pocket enclosed between two contactless rough

243

surfaces of infinite extent. The contacts between these two surfaces can in principle mod-

244

ify the geometry of the resulting dissolution surface. However, they are assumed to be

245

sufficiently loose in a stylolite and thus the main morphological results are not affected.

246

This assumption simplifies the problem since the dissolution process, happening on both

247

sides of the stylolite (Figure 1), can be described as the dissolution of a solid half-plane in

248

contact with a fluid. With this geometry, the small and large scale self-affine behaviors of

249

the dissolution surface and the associated roughness exponents (or Hurst exponents) are

250

well reproduced. The model leads to the characteristic exponents typically observed in

251

previous studies [Renard et al., 2004; Schmittbuhl et al., 2004; Brouste et al., 2007; Ebner

252

et al., 2009b].

253

The average stylolitic plane is defined along the x and y-axis (Figure 2). To have

254

better statistics on the morphology of the studied surfaces, the model is assumed to be

255

invariant by translation along the y-axis. It allows to us to describe a larger range of

256

scales at the same numerical cost and to numerically solve the self-affine behavior of the

257

resulting pressure-solution surface over a larger number of orders of length scales. The

258

same approach can be considered using invariance by translation along the x-axis. In

259

the model we assume a mechanical equilibrium throughout the system and express the

260

dissolution rate as a function of the stress tensor and of the area of interface per unit

261

volume.

262

2.1. Force perturbation related to the mechanical equilibrium along the fluid- solid interface

First, we express the mechanical equilibrium at the solid-fluid interface (Figure 2).

263

The convention adopted is that compressive stresses and compactive strains are negative

264

Landau and Lifchitz [1986]. The far-field stress applied to the host rock is denoted by

265

¯¯

σ

. The largest principal stress axis, perpendicular to the average plane of the stylolite,

266

is defined along the z-axis. The fluid pocket transmits all the load through itself (The

267

boundary condition of the fluid pocket is approximated as undrained for that respect:

268

if there is any flow, from or into the fluid pocket, it happens slowly, via the lateral

269

ends. If there is any contact between the opposite walls perpendicular to the main fluid

270

direction, the load transmitted through this contact is neglected). The fluid pressure

271

is thus homogeneous and equal to the largest principal stress applied to the host rock,

272

considering the integral of the local stress field ¯¯ σ along an elongated rectangular boundary

273

(dashed line in Figure 1):

274

p = − σ

(1)

275

Locally, the local stress ¯¯ σ is split between the far-field asymptotic value ¯¯ σ

and a pertur-

276

bation generated by the irregular nature of the interface ¯¯ σ

:

277

σ(x) = ¯¯ ¯¯ σ

+ ¯¯ σ

(x) (2)

278

The far-field stress unit vectors ˆ x and ˆ z along the x and z-axis are assumed to be the

279

principal directions i.e.

280

¯¯

σ

= σ

xˆ ˆ x + σ

z ˆ z ˆ (3)

281

Here, the notations ˆ xˆ x and ˆ z z ˆ correspond to unit matrixes composed from the unit

282

vectors, as e.g. are ˆ y y, ˆ ˆ xˆ y, or ˆ z x. This canonical basis for the matrixes is composed ˆ

283

from the doublets of unit vectors ˆ x, y ˆ and ˆ z. For example, ˆ xˆ z represents the unit matrix

284

with all components equal to zero, apart from a unit in the lign corresponding to the x

285

coordinate, and the column corresponding to the z one, so that for a pair of vectors u, v

286

applied to the left and right of this matrix, u · (ˆ xˆ z) · v = (u · x)(ˆ ˆ z · v) = u

v

. In other

287

terms, with cartesian components along directions of indexes i and j, and the help of the

288

Kronecker symbol δ, the components of the matrix ˆ x z, for example, are: (ˆ ˆ xˆ z)

= δ

δ

.

289

This convention to define the canonical basis of matrix space (nine elementary second

290

order dyadic products like ˆ xˆ z) from the three basic unitary vectors of the vectorial space,

291

ˆ

x, y ˆ and ˆ z is, for example, defined by [Gonzalez and Stuart, 2008].

292

For a stylolite, the largest compressive stress axis is normal to its average plane and

293

thus to the average fluid pocket direction:

294

| σ

| > | σ

| (4)

295

This relation has strong implications on the stability of the surface pattern emerging from

296

the dissolution process. The far-field deviatoric stress is defined as:

297

σ

= ( | σ

| − | σ

| ) = (σ

− σ

) (5)

298

To express the force perturbation related to the curved nature of the interface, we define

299

the unit vector ˆ n normal to the surface pointing towards the fluid. This vector is assumed

300

to be close to the principal stress axis. In the following, we will consider small-angle devia-

301

tions from a straight surface, and the results will therefore be valid for small surface slopes

302

only. The model presented below aims to describe the onset of the stylolite propagation

303

from a flat surface, and it will also describe the evolution of large wavelength modes, if

304

the aspect ratio of such modes (ratio of the amplitude over the wavelength) stays small,

305

corresponding to small effective slopes at large wavelength.

306

The interface is described as a single-valued function z(x) and the slopes are assumed to

307

be of the order i.e. that | ∂

(z) | ∈ O() 1. Since the normal ˆ n to the interface of slope

308

∂

z can be expressed by the conditions of normality to the interface, ˆ n · (1, ∂

z)

= 0

309

(at any order or ), and by its unitary norm ˆ n

= 1, it is in general ˆ n = ( − (∂

z)ˆ x +

310

ˆ

z)/

1 + (∂

z)

. Using the above limit of small slopes, developing in , we obtain to

311

leading order

312

ˆ

n = ˆ z − (∂

z)ˆ x + O(

) (6)

313

(The order O() is absent from ˆ n).

314

The local mechanical equilibrium at the solid-fluid interface is expressed as:

315

σ · ˆ n = − pˆ n (7)

316

And with equations (1-7) the force perturbation (illustrated in Figure 3) becomes:

317

δf (x) = σ

(x) · ˆ n =

318

− pˆ n − σ

· n ˆ

319

= σ

[ˆ z − (∂

z)ˆ x] − σ

z ˆ + σ

(∂

z)ˆ x

320

= (σ

− σ

)(∂

z)ˆ x

321

= σ

(∂

z)ˆ x (8)

322

2.2. Chemico-mechanical coupling

Next, we express the chemico-mechanical coupling. The dissolution speed normal to the

323

solid/fluid interface (in mol.m

.s

), is to the first order proportional to the chemical

324

potential ∆µ of the chemical product dissolving [Kassner et al., 2001; Misbah et al., 2004;

325

Schmittbuhl et al., 2004; Koehn et al., 2007]:

326

v = m∆µ (9)

327

where

328

m = k

Ω/RT (10)

329

is the mobility of the dissolving species, R = 8.31 J.mol

.K

is the universal gas

330

constant, T is the temperature in Kelvin, k

is a dissolution rate which can be measured

331

experimentally, and Ω is a molar volume. For calcite, Ω ' 4 · 10

m

.mol

and k

' 10

332

mol.m

.s

for dissolution in water at atmospheric pressure and 298

K [De Giudici , 2002;

333

Schmittbuhl et al., 2004]. The difference in chemical potential from the solid state to the

334

fluid state is [Kassner et al., 2001; Misbah et al., 2004; Koehn et al., 2007]:

335

∆µ = ∆Ψ

+ Ω∆P

+ Ωγκ (11)

336

Considering a solid state at given pressure and elastic free energy in chemical equilibrium

337

with the fluid, ∆Ψ

and ∆P

are defined respectively as the change in Helmoltz free energy

338

per mole and the change in stress normal to the interface. The last term corresponds to the

339

surface energy with κ = ∂

z, the surface curvature (the inverse of the radius curvature)

340

and γ the surface tension between the solid and the fluid phase. In a particular case,

341

neglecting temperature variation effects and assuming that the fluid composition is in

342

chemical equilibrium with a solid flat surface at normal pressure p and stress σ

, Eq.(11)

343

reduces to :

344

∆µ = 0 (12)

345

κ = 0 (13)

346

More generally, by definition ([Kassner et al., 2001]:

347

∆Ψ

+ Ω∆P

= Ω∆u

, (14)

348

where

349

∆u

= u

(σ) − u

(15)

350

and

351

u

= [(1 + ν)σ

σ

− νσ

σ

]/4E (16)

352

is the elastic free energy per unit volume with E the Young’s modulus and ν the Poisson’s

353

ratio of the elastic solid [Kassner et al., 2001; Landau and Lifchitz , 1986].

354

To take into account the dissolution speed variations associated to the morphology of

355

the stylolite, we develop the dissolution speed to the leading order as:

356

v = v

+ v

(17)

357

With equations (9-16):

358

v

= k

Ω

RT

[(1 + ν)σ

σ

− νσ

σ

]

4E − u

!

359

= k

Ω

RT E (αp

− α

p

) (18)

360

The geometrical factor α is computed assuming σ

= σ

= − p

+ σ

/3 and σ

=

361

− p

− 2σ

/3:

362

α = 9(1 − 2ν) + 2(1 + ν)σ

/p

12 (19)

363

α

is expressed with a similar expression and characterizes the chemical equilibrium with

364

the fluid at the referential state as a function of the pressure p

and the shear stress

365

σ

. Typically, for a limestone with a Young’s modulus E = 80 GPa stressed at p

' 10

366

MPa (which corresponds to a few hundred of meters deep in sedimentary rocks) and for a

367

fluid with a chemical composition in equilibrium with the solid, the dissolution speed at

368

the solid-fluid interface in a limestone is of the order of:

369

v

' 10

to 10

m.year

370

2.3. Consequences for the stability of the dissolution process

From the local mechanical equilibrium and the nature of the chemico-mechanical cou-

371

pling, some important considerations can be inferred about the morphological stability of

372

the dissolution surfaces. This behavior depends on the orientation of the surfaces with

373

respect to the far-field stress.

374

Previously we have shown how to express the force perturbation arising from the mis-

375

match between the solid-fluid interface orientation and the principal axis of the far-field

376

stress tensor ˆ x (equation 8).

377

This relationship holds independently of the relative magnitudes of the principal stresses

378

σ

and σ

. If the largest principal stress is tangential to the interface, which is not the

379

case for stylolites, σ

< 0 and the sign of δf (x) · x ˆ is opposite to the slope of the interface

380

∂

z. Such tangential force perturbation is concentrated at the points lying ahead of the

381

average dissolution front (Figure 4). The elastic forces concentrate stress at the valleys of

382

the dissolution front where the free energy is thus higher. This leads to an increased dis-

383

solution speed at the points lying ahead of the averaged front. The dissolution propagates

384

downwards. The points at the crests, i.e. located behind the averaged dissolution front,

385

show a reduced rate of dissolution thus pushing them further from the average front. The

386

points lying out of the average dissolution plane tend therefore to depart further from

387

the average position. The elastic force is in this situation a destabilizing force. On the

388

contrary, the surface tension tends to stabilize the process by decreasing the surface area

389

by flattening the interface.

390

The competition between the elastic long-range destabilizing forces and the surface

391

tension short-range stabilizing forces leads to the ATG interface instability. The fastest

392

growing wavelength is determined by the balance between these long-range destabilizing

393

and short-range stabilizing effects. Such instability arising in stressed solids was studied

394

theoretically [Asaro and Tiller , 1972; Grinfeld , 1986; Misbah et al., 2004] and observed

395

experimentally in stressed soluble crystals immersed in a saturated fluid [Den Brok and

396

Morel , 2001; Koehn et al., 2004].

397

If the largest principal stress lies perpendicular to the interface, as for stylolites, σ

> 0

398

and the sign of δf (x) · x ˆ is the same as the slope of the interface ∂

z. Such tangential

399

force perturbation is concentrated at the points lying behind of the average dissolution

400

front (Figure 5). The elastic forces concentrate stress at the crests of the dissolution front

401

where the free energy is thus higher. This leads to an increased dissolution speed for

402

the points lying behind the averaged front. The dissolution propagates downwards. The

403

points at the valleys, i.e. located ahead of the averaged dissolution front, tend to come

404

back to the average position. The elastic force is a stabilizing force in this situation. Here,

405

the surface tension is again a stabilizing process.

406

Since the long-range elastic force and the short-range surface tension force are stabilizing

407

forces, if the modeled solid properties are purely homogeneous (i.e. homogeneous elastic

408

solid with homogeneous dissolution rate properties), the model predicts the flattening of

409

any initial non-plane surface with time.

410

Consequently, to model the morphogenesis of stylolites, which are rough surfaces, we

411

will take here into account the disorder linked to the material properties.

412

2.3.1. Consequence on initial evolution of trapped fluid pocket

413

In summary, the above arguments show that an elementary bump of a flat surface dis-

414

appears for σ

> 0, or grows for σ

< 0. Qualitatively, if the argument on the stability

415

of surfaces depending on their orientation on the principal stress axis extends for more

416

local orientations along trapped fluid pockets, one should observe the following: for the

417

sides of a fluid pocket lying tangentially to the largest stress, these should develop instable

418

grooves penetrating into the solid, similarly to the ATG instability case. On the contrary,

419

the sides normal to the largest stress direction should remain relatively flat, apart from

420

the fluctuations due to the disorder. These small variations along the surfaces normal to

421

the principal stress axis, and the penetrations of grooves of characteristic wavelength in

422

the rock along the direction of the weakest stress, should lead to the development of elon-

423

gated structures, and merge initially separated fluid pockets (or clay-enriched pockets).

424

This qualitative mechanism is illustrated on Figure 6. This expectation of qualitative

425

evolution is indeed compatible with the mechanism of development of anti-cracks numer-

426

ically obtained by Koehn et al. [2003]. The experimental grooves observed along the free

427

surface on the sides of a fluid-filled cylindrical pocket by Den Brok and Morel [2001] also

428

displayed this trend.

429

2.4. Expression of the dissolution speed perturbation as a function of the interface shape

To model the disorder in the solid we assume that the material properties (related to the

430

solid grains) vary in a random and spatially uncorrelated way. This disorder can originate

431

from the diversity of grain composition, grain size or orientation, i.e. it represents the

432

small scale heterogeneities present in the rock For example, the dissolution rate k can be

433

expressed as an averaged term k

plus some spatial variations of zero average η(x, z) · k

:

434

k = k

(1 + η(x, z(x))) (20)

435

The random variable η is a quenched disorder with no spatial correlations and is charac-

436

terized by its mean < η >= 0 and its variance < η

> assumed to be small enough to

437

keep small local slopes. The dynamics of the dissolving interface z(x, t) can be expressed

438

from equations (9-17) as:

439

v = − ∂

z = kΩ

RT (∆u

+ γ∂

z),

440

= k

Ω

RT (1 + η) { (1 + ν)[(σ

+ σ

)(σ

+ σ

) − ν(σ

+ σ

)

]/4E − u

+ γ∂

z }

441

= k

Ω

RT { (1 + ν)[σ

σ

− ν(σ

)

]/4E − u

}

442

+ k

Ω

RT η { (1 + ν)[σ

σ

− ν(σ

)

]/4E − u

}

443

+ k

Ω

RT { (1 + ν)[2σ

σ

− 2ν(σ

σ

)]/4E + γ∂

z } (21)

444

i.e., using Eq.(18) for the expression of σ

σ

− ν(σ

)

, a dissolution speed separated

445

between an average homogeneous speed v

and a leading order of the perturbations v

,

446

first order in as

447

∂

z(x, t) = − v

− v

(x, t) (22)

448

with v

the dissolution speed given by equation (18) and v

a deviation of the dissolution

449

speed with respect to the average dissolution speed v

expressed as:

450

v

= k

Ω

RT

(αp

− α

p

)

E η(x, z(x))

451

+ k

Ω

RT γ∂

z(x)

452

+ k

Ω

RT

[(1 + ν)σ

σ

− νσ

σ

] 2E

!

(23)

453

σ

is the stress perturbation mentioned previously in equation (2). It is generated by the

454

surface distribution of the tangential force perturbation δf (x) due to the irregular nature

455

of the interface.

456

The first term is a quenched disorder term leading to the roughening of the interface.

457

The second one is a stabilizing quadratic short-range term arising from the surface tension.

458

The last term can be expressed via a non-local kernel from the shape of the interface z(x)

459

by integrating the elastostatic equations in the solid half-plane.

460

2.5. Detailed form of the elastic long-range interaction kernel

The stress perturbation induced by the force perturbation δf (x) (equation 8) exerted

461

on the surface can be determined via the Green function method. Following Landau and

462

Lifchitz [1986], the displacement induced by an elementary force ˆ x applied at the origin

463

(0, 0, 0) on a semi-infinite solid is:

464

a

(x, y, z) = 1 + ν 2πE

(

2(1 − ν)r + z

r(r + z) + (2r(νr + z) + z

r

(r + z)

x

)

465

a

(x, y, z) = 1 + ν 2πE

(

2r(νr + z) + z

r

(r + z)

xy

)

466

a

(x, y, z) = 1 + ν 2πE

(

(1 − 2ν)x r(r + z) + zx

r

)

(24)

467

where r is the distance relatively to the force application point at (0, 0, 0), i.e. r

=

468

x

+ y

+ z

. The associated strain applied on the solid is:

469

= 1

2 (∂

a

+ ∂

a

) (25)

470

and the associated stress is:

471

f

(x, y, z) = E

1 + ν (

+ ν

1 − 2ν

δ

) (26)

472

The stress associated to the point force ˆ x applied on the surface of normal ˆ z at the origin

473

is equal at the origin itself to ˆ xˆ z + ˆ z x. ˆ

474

Since the model treated here is invariant by translation along y, the force perturbation δf (u) = σ

(∂

z)(u)ˆ x is exerted at any v ∈ ] −∞ , ∞ [ and the resulting displacement field at (x, y, z), is solely dependent on (x, z) and can be expressed, by linearity of the elastostatics equations, (similarly to the elastostatic Green function method detailed in Eq.(8.14) by [Landau and Lifchitz , 1986]), as a displacement field w of components

w

(x, y = 0, z) =

Z ∞

u=−∞

Z ∞

v=−∞

a

(x − u, − v, z)dudv δf (u).ˆ x (27)

The associated strain perturbation is

= 1

2 (∂

w

+ ∂

w

), (28)

and the associated stress,

σ

(x) = E

1 + ν (

+ ν

1 − 2ν

δ

) + δf (x)(δ

δ

+ δ

δ

)δ(z), (29)

where the first term represents the stress induced by the elastic deformation, and the

second one the direct application of the force perturbation on the surface. In the above,

the spatial derivative of Eq.(28) can be exchanged with the integration in Eq.(27), to

obtain

(x, y = 0, z) =

Z ∞

u=−∞

Z ∞

v=−∞

(x − u, − v, z)dudv δf (u).ˆ x. (30)

Recalling the expression of the force perturbation, Eq.(8), from Eq.(29), the stress per-

475

turbation along the surface, at z = 0, is thus:

476

σ

(x) = σ

· p.p.[

Z ∞

u=−∞

du (∂

z)(u)

477

∗

Z ^∞

v=−∞

f

(x − u, − v, 0)dv]

478

+σ

(∂

z)(x)(δ

δ

+ δ

δ

) (31)

479

where p.p. refers to the principal part of the integral. Taking the derivatives of the

480

displacement field (equation 25), we can calculate the associated stress. Integrating this

481

result along the y-axis gives:

482

Z ∞

v=−∞

f

(x, − v, 0)dv = − 2ν

πx (δ

δ

+ δ

δ

) (32)

483

and thus,

484

σ

(x) = − 2νσ

π · p.p.[

Z ∞

x⁰=−∞

du (∂

z)(u)

x − u ](ˆ xˆ x + ˆ y y) ˆ

485

+σ

· (∂

z)(x)(ˆ x z ˆ + ˆ z x) ˆ (33)

486

The elastic energy perturbation associated to the interface deformation can be computed

487

using equation (33) and the relation:

488

σ

= − (p

− σ

/3)(ˆ xˆ x + ˆ yˆ y) − (p

+ 2σ

/3)ˆ z z ˆ (34)

489

It results in:

490

u

= [(1 + ν)σ

σ

− νσ

σ

]

491

2E

= 2ν[(1 − 2ν)p

]

πE ∗ σ

· p.p.[

Z ^∞

u=−∞

du (∂

z)(u)

x − u ] (35)

492