doi:10.1144/GSL.SP.2005.243.01.03 2005; v. 243; p. 11-24 Geological Society, London, Special Publications

 Daniel Koehn, Jochen Arnold and Cees W. Passchier  

numerical studyFracture and vein patterns as indicators of deformation history: a 

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Fracture and vein patterns as indicators of deformation history: a numerical study

DANIEL KOEHN, JOCHEN A R N O L D & CEES W. P A S S C H I E R

Tectonophysics, Institute for Geosciences, Becherweg 21,

University of Mainz, 55099 Mainz, Germany

(e-mail: koehn @ mail. uni-mainz, de)

Abstract: Fracture and vein patterns in the brittle crust of the Earth contain information on the stress and strain field during deformation. Natural examples of fracture and vein patterns can have complex geometries including combinations of extension and conjugate shear frac- tures. Examples are conjugate joint systems that are oriented with a small angle to the prin- cipal stress axis and veins that show an oblique opening direction. We developed a discrete numerical model within the modelling environment 'Elle' to study the progressive develop- ment of fractures in two dimensions. Results show that pure shear deformation alone can produce complex patterns with combinations of extension and shear fractures. These pat- terns change in geometry and spacing depending on the Young's modulus of the deforming aggregate and the initial noise in the system. A complex deformation history, including primary uniaxial loading of the aggregate that is followed by tectonic strain, leads to con- jugate shear fractures. During progressive deformation these conjugate shear fractures may accommodate extensional strain or may be followed by a secondary set of extension fractures. The numerical patterns are consistent with joint, fault and vein geometries found in natural examples. The study suggests that fracture patterns can record complex deformation histories that include primary uniaxial loading due to an overlying rock sequence followed by tectonic strain.

Brittle deformation involving fracturing of rocks is a very important deformation mechanism in the upper crust of the Earth (Patterson 1978; Ranalli 1995). Many sedimentary rocks that are exposed today are filled with joints, faults and veins with characteristic patterns reflecting the importance of brittle deformation (Price & Cosgrove 1990). In addition, fracture and vein patterns are used by structural geologists as a map of the far-field stress orientation at the time of fracture formation (Ramsay & Huber 1983; Price & Cosgrove 1990; Oliver & Bons 2001).

Fractures are generally classified into mode I, mode II and mode III fractures (Pollard & Segall 1987; Scholz 2002). Mode I fractures are exten- sional and open perpendicular to the maximum tensile stress. They accommodate strain by opening and may contain vein material that pre- cipitates in the open space. Mode II fractures are shear fractures that develop at an angle to the maximum principal stress. They accommodate strain by slip along the fracture plane and are therefore called shear fractures or faults if signifi- cant slip has taken place along them (Bonnet et al. 2001). Mode III fractures are important in

three dimensions and will not be considered further in this paper since the presented models are two-dimensional. In order to understand what kind of fracture will form under a given stress condition and what orientation it will have with respect to one of the major stress axes it is useful to construct a Mohr circle diagram (Jaeger & Cook 1976) with the normal stress on the x-axis and the shear stress on the y-axis (Fig. 1). The Mohr envelope or failure curve in Figure 1 is constructed after Griffith (Griffith 1920; Jaeger & Cook 1976; Scholz 2002). If the Mohr circle crosses the failure curve in the regime of tensile stress, mode I frac- tures will develop, if it cuts the failure curve in the compressive stress regime, mode II fractures will form with an angle 0 to the direction of the compressive stress. We will term fractures that form by a coalescence of small mode I cracks, extension fractures (joints in a larger volume of rock or veins if they are filled with new material) and fractures that form by a coalescence of mainly mode II cracks, shear fractures (faults in a larger volume of rock if slip takes place). In addition, Hancock (1985) and Price & Cosgrove (1990) mention an intermediate

From: GAPAIS, D., BRUN, J. P. & COBBOLD, P. R. (eds) 2005. Deformation Mechanisms, Rheology and Tectonics: from Minerals to the Lithosphere. Geological Society, London, Special Publications, 243, 11-24. 0305-8719/05/$15.00 �9 The Geological Society of London 2005.

12 D. KOEHN ET AL.

hybrid i ~ mode II

(3"1 , Oli ~ failure curve Ze=4To(O'n+To)

i , ,, ~ , T

" , , ..,

", 0- I T (In

', ~ failure ' AO curve

Fig. 1. A Mohr circle diagram with the normal stress (o-.) on the x-axis and the shear stress (~-) on the y-axis. The failure curve is constructed using the Griffith failure criterion (Jaeger & Cook 1976). Two Mohr circles with different sizes are shown. The smaller circle cuts the failure envelope in the tensile regime and produces mode I fractures and hybrid extension/shear fractures. The larger circle cuts the failure envelope in the compressive regime and produces mode II shear fractures. The mean stress (o-m), differential stress (Act) and the two main principal stress components (o'l, o'2) are indicated next to the larger Mohr circle. To represents the tensile strength.

fracture type called hybrid extension/shear frac- tures that form at an angle to the compressive stress direction but still lie in the tensile stress regime (Fig. 1) and form probably by a coalesc- ence of mode I and II cracks. One has to note, however, that the occurrence of hybrid exten- sion/shear fractures as propagating cracks and their orientation with respect to the principal stress direction and thus their connection to the parabolic failure curve in Figure 1 is question- able (Engelder 1999; Ramsay & Chester 2004).

Rocks in the brittle regime can only sustain small amounts of elastic strain before they start to form fractures (Means 1976). If fractures develop, the overall behaviour of the rock will be plastic (Paterson 1978). Fractures accommodate strain in a rock and relieve the overall stress. However, since fractures, are two-dimensional planar structures, in order to accommodate three- dimensional strain, complex fracture patterns develop. Anderson (1951) introduced the idea of conjugate faults, which intersect in the intermedi- ate stress direction. The plane perpendicular to the intermediate stress axis then includes the minimum and maximum stress directions and the slip vectors of the faults. This concept, however, only works if the intermediate stress is zero or

close to zero. Otherwise more complex fault or fracture patterns develop with at least four sets of coeval faults, arranged in an orthorhombic symmetry (Oertel 1965; Aydin & Reches 1982; Krantz 1988).

Fractures that are filled with vein material pose a problem. Rocks are normally loaded by the overlying rock sequence before they are deformed tectonically, so both the vertical and horizontal stresses will be compressional. There- fore extension fractures are not expected to form since the Mohr circle will cut the failure envel- ope in the compressive regime (Jaeger & Cook 1976; Means 1976; Suppe 1985; Price & Cosgrove 1990). However, veins are formed quite often in the upper crust, so an alternative mechanism is needed to produce tensile stresses. Two possible sources are (1) a heterogeneous layered sequence where tensile stresses are con- centrated in competent layers forming fracture- boudinage or (2) the presence of a high fluid pressure that shifts the Mohr circle towards the tensile regime (Price & Cosgrove 1990; Ranalli 1995). In these cases one can find extension frac- tures that are filled with vein material if material can precipitate. An example of a hybrid shear fracture that is filled with vein material and later- ally extends into extension fractures is shown in Figure 2a. An interesting question is if a vein set like the one shown in Figure 2b can also be associated with a single set of conjugate hybrid shear fractures. Examples of conjugate hybrid joint sets can be found in Hancock (1985).

In this paper we model the dynamic develop- ment of fracture patterns in a two-dimensional approach. We simulate different deformation histories starting from a simple case where we investigate different internal model parameters. Then we extend to more complex deformation histories that include gravitational loading and tectonic strain. We will argue that the initial fracture geometries that later develop into veins contain a record of the gravitational loading that preceded tectonic deformation. It will be shown that even during a simple deformation history different stress states in the samples will lead to different kinds of failures.

The model

In order to model two-dimensional fracture patterns we developed a discrete-element model based on the work of Malthe-Screnssen et al. (1998b) and combined it with the modelling environment 'Elle' (Jessell et al. 2001). With this kind of model, linear elastic behaviour of rock aggregates can be simulated with the full description of the strain and stress field. The

(a) FRACTURE AND VEIN PATTERNS

(b)

13

Fig. 2. (a) Sketch of a vein that may have developed partly as a hybrid shear fracture (after Price & Cosgrove 1990, Fig. 1.63). (b) Bedding plane with conjugate sets of veins from a locality near Sestri Levante in the Liguride units, Italy (see hammer for scale).

model also allows the introduction of fractures by bond-breaking and can thus be used to inves- tigate the behaviour of the aggregate beyond linear elasticity theory. Studies with this kind of model have shown that it reproduces the scaling behaviour and statistics of experimental fracture patterns (Walmann et al. 1996; Malthe- S0renssen et al. 1998a, b, 1999) and more com- plex systems (Jamtveit et al. 2000; Flekk0y et al. 2002). In the discrete model circular particles are connected with their neighbours by linear elastic springs (Fig. 3a). We use a triangular lattice in two dimensions since it reproduces linear elas- ticity theory with only nearest neighbour inter- action (Flekkcy et al. 2002). The force acting on a particle from a neighbour is proportional to the extension or compression of the connect- ing spring with respect to the equilibrium dis- tance. By definition, compressive stresses in the model are negative and tensile stresses positive. The lattice is in equilibrium condition if all forces acting on a particle cancel out. If the equi- librium condition is perturbed, for example by applying a deformation to the lattice, particles

. ( ~ ) ~ ~ ~ (b bound,ary (c) fracture

Fig. 3. Sketches of the discrete particle model that is used for the numerical simulations. (a) Triangular lattice showing particles and springs that connect particles. (b) Boundary particles (in grey) are used to apply deformation on the modelled aggregate. (c) Fractures are induced by the possibility of breaking springs. Springs break once they have reached a critical tensile stress and are irreversibly removed from the model.

are moved according to resulting forces until all particles are in an equilibrium position defined by a given threshold. In order to find equilibrium for the whole lattice efficiently it is over-relaxed by moving the particles by an 'over-relaxation factor' beyond their equilibrium condition (Allen 1954). In the model, particles fill an initial area called the deformation box. Particles along the box boundaries are defined as wall-particles and are used to apply kinematic boundary conditions (Fig. 3b). These particles are fixed perpendicular to the boundary of the model but can move freely parallel to it, for example, free-slip boundary conditions are applied. In order to strain the model, walls are moved inwards to apply compression or out- wards to apply extension. Once a wall is moved all particles in the deformation box are moved assuming homogeneous deformation. Then the relaxation algorithm starts. Once the lattice is in an equilibrium condition after a deformation step, all springs are checked for breaking. If the stress associated with a spring reaches a critical tensile value, fracturing is induced by breaking springs (Fig. 3c). The spring with the highest probability of breaking will do so and a new relaxation routine will start. This procedure is repeated until all springs are below the breaking threshold and a new deformation step is applied. The fracturing process is therefore highly non- linear; the whole lattice can fail at once between two deformation steps. This micro- scopic failure criterion will result in shear and extension failure between two neighbouring par- ticles because particles are connected with six springs to their neighbours. Particles with broken springs are still repulsive.

The coupling of the discrete code with the 'Elle' environment is performed as follows. In 'Elle' the microstructure of a rock is defined by nodes, which are connected with boundary

14 D. KOEHN ETAL.

Fig. 4. (a) Combination of the 'Elle' modelling environment with the discrete model. In Elle, grains are defined by double and triple nodes that are connected by straight segments. Particles of the discrete model that lie within an Elle grain have the same properties. (b) Example of a microstructure that was used for most of the simulations shown in this paper.

segments that define grains (Fig. 4a). In the dis- crete code grains are filled with particles. All par- ticles within a grain can have specific parameters like the elastic constant and the breaking strength of springs. Springs that connect particles of dif- ferent grains define grain boundaries and have different properties to springs that lie within grains. In all the presented models grain boundary springs have about half the tensile strength of intragrain springs. This value is chosen in order to have an influence of grain shapes on fracture development. Therefore most fractures will be intergranular. Intragranular fractures mainly appear when shear fractures develop. We expect grain boundaries to have a lower cohesion due to the presence of fluids or impurities.

In order to reduce lattice effects, noise is dis- tributed in the model by three different methods. First an initial microstructure is drawn and used as an input file with a certain spatial distribution of grain boundaries (Fig. 4b). Then a statistical distribution is applied to the elastic constants of different grains. Each grain has an individual elastic constant chosen by a random function from a gauss distribution. The mean elastic con- stant of the whole deformation box is represented by the mean value of the distribution. The break- ing strength of all springs is statistically dispersed using a linear distribution because a rock is also expected to be disordered on a scale smaller than the grain size (Malthe-SOrenssen et al. 1998b). This distribution is not dependent on grains and grain boundary springs still have about half the breaking strength of intragrain springs.

The presented simulations have a resolution of 184 800 particles within the deformation box. In all simulations we strain the model in small steps of 0.02%. In the model the unscaled mean value of the initial elastic constant is 1.0 and the breaking strength of intragranular springs is 0.006. Grain boundaries always have half the breaking strength of intragranular springs. We scale these values in the results section in order to discuss the models in a geological context. We assume that the breaking strength of grain boundaries is 30MPa. Therefore a value of 0.0001 in the model scales to 1 MPa. An initial spring constant of 1.0 thus represents an elastic constant of 10 GPa. This leads to a mean tensile strength of the aggregates used in the simulations of about 25 MPa and a mean compressive strength of about 90 MPa under pure shear deformation (starting from zero strain and zero stress). This compressive strength is relatively low because in pure shear deformation one principal stress is tensile if the experiment is not preloaded. If this stress is also compressive the breaking strength will be considerably higher. This relation is also seen in laboratory experiments and is predicted by theory (Jaeger & Cook 1976; Paterson 1978). If the aggregate in our simulations is preloaded before tectonic deformation is applied (the con- fining pressure is increased), the compressive strength rises to about 140-190 MPa.

Results

We present a number of simulations of fracture development with different deformation histories

FRACTURE AND VEIN PATTERNS 15

and show the dependence of the pattern on the initial noise in the system and differences in the mean elastic constant of the aggregate. First we discuss fracture patterns that develop under pure shear deformation. We investigate the influ- ence of noise and mean elastic constant of the aggregate on pattern formation using examples that experience the same pure shear deformation history. Then we move on to discuss deformation patterns that develop during area increase (expanding the model area). Finally we include complex deformation histories where samples experience an initial uniaxial loading and then a tectonic deformation. This leads to a number of deformation histories, which produce the vein and fracture patterns geologists can find in some field areas.

Pure shear deformation

First we examine fracture patterns that are gener- ated in a sample that experiences pure shear deformation. The initial sample is not stressed, deformation starts from a completely relaxed state. Figure 5a shows the stress-strain relation- ship during deformation. Different curves corre- spond to the maximum and minimum stress, the

mean stress and the differential stress. Note that compressive stress is negative. Figure 5b shows schematic Mohr circles that represent stress states for some of the deformation events shown in Figure 5a. The developing patterns are illustrated in Figure 5c. Starting from a com- pletely relaxed state, during progressive defor- mation the stress field behaves according to linear elasticity theory. Principal stresses are oriented parallel to the directions of compression and extension and increase linearly with the same slope into the tensile and compressive field. Con- sequently the mean stress remains zero and the differential stress curve has a slope that is twice as steep as that of a single principal stress com- ponent (Fig. 5a). The corresponding Mohr circle grows but remains stationary at the inter- section of the axis of the diagram. A peak in the curve of the tensile stress marks the intersec- tion when the Mohr circle reaches the tensile part of the failure envelope. At this point tensile frac- tures grow in the sample perpendicular to the extension direction. They cross most of the sample and show a distinct spacing. These frac- tures accommodate extensional strain and relax tensile stresses. This effect is also seen in the stress-strain curve in Figure 5a. The tensile

~(uPa)

(a) 8o (b)~l;~n 400 A(~~ (~m -40 I (+) (') ~ /

0 0.01 0.0Z E ~ A ~ T

- , ~ "

Fig. 5. Simulation of pure shear deformation of an aggregate. (a) Stress-strain relation showing the two principal stresses (oh, ~r2), the mean stress (Crm) and the differential stress (Ao-). (b) Schematic Mohr circle diagram of (a) showing Mohr circles during different stages of the simulation. They mark the initial starting configuration, mode I failure and mode II failure. Note that the x-axis is negative towards the right-hand side since compressive stresses are negative in this paper. (c) Three successive stages during the simulation. The picture on the left-hand side shows development of mode I fractures. The pictures in the middle and on the right-hand side show development of mode II shear fractures.

16 D. KOEHN ET AL.

stress drops, the mean stress starts to increase and deviates from zero and the differential stress has a slope that is less steep. The compressive stress is not much affected by extension fractures because the vertical load is still supported by the sample. The Mohr circle in Figure 5b starts to move from the tensile regime into the com- pressive region and continues to grow in radius. The next failure event is reached when the differential stress is high enough so that the Mohr circle cuts the upper part of the failure curve and shear fractures develop. Before this moment is reached the stress curves show increasingly flattening slopes due to the local development of small shear fractures prior to failure of the whole sample. The failure of the sample is accommodated by slip along shear fractures (shear fracture becomes a distinct fault), which relaxes the compressive stress in the sample in a number of steps as seen in the stress-strain curve. The differential stress is also relaxed until the sample reaches a quasi steady-state where stresses remain almost con- stant and the bulk behaviour is almost purely plastic.

This deformation history will result in the primary growth of extension fractures and a secondary growth of conjugate shear fractures.

Extension fractures will progressively open to accommodate extensional strain so that veins can form whereas the shear fractures will mainly show slip along their surfaces and develop into faults. Thus, in a natural example one would expect only one set of veins that develops out of extension fractures.

Differences in initial noise

and elastic constant

Differences in initial statistical noise and in the mean elastic constant of the aggregate influence the developing pattern. By initial noise in the system we mean the width of the distribution of elastic constants of different grains and the breaking strength of individual grains. Figure 6 shows two simulations with identical initial microstructure and elastic constants as in Figure 5, but with varying width of the distri- bution of the breaking strengths. The simulation shown in Figure 6a has a distribution of break- ing strengths from 24 to 36 MPa whereas the simulation shown in Figure 6b has a distribution of breaking strengths ranging from 12 to 48 MPa. For each simulation two successive stages at the same amount of strain are shown, deformation is

(a} t ) i

,.

/ ( ?

/

... ,~ : ~,

(b) ./

3,

4 .

�9 '!

Fig. 6. Two simulations with the same boundary conditions but different initial distributions of breaking thresholds, where (a) has a narrower distribution of breaking thresholds than (b). (a) Less noise in the system produces stronger localization of structures. (b) More noise produces more dispersed fractures with different sizes.

FRACTURE AND VEIN PATTERNS 17

pure shear where compression is vertical. Stage one shows the initial development of extension fractures and stage two the development of secondary shear fractures. Two important differ- ences can be observed between the different distributions of breaking strengths. First a wider distribution of breaking strengths results in a more dispersed development of fractures whereas a narrow distribution results in localized fractures that cut the whole aggregate. Secondly, the absolute breaking strength is lowered using a wider distribution of breaking strengths; more fractures develop in Figure 6b than in Figure 6a. The first effect can be explained as follows. A narrow distribution of breaking strength, which means that the initial noise in the system is low, represents a very brittle material. A lot of springs in the aggregate will reach their tensile strength within a single defor- mation step. This will result in an almost instan- taneous propagation of large fractures through the aggregate. In particular, an extension fracture will continue to propagate once it nucleates since tensile stresses at the fracture tip will increase while the fracture grows in length. Once a large fracture is present it will be able to relax stresses

in the surrounding aggregate so that additional fractures will only develop at a certain distance to the initial fracture. This stress shielding mech- anism produces a distinct spacing of fractures in the aggregate. If the width of the distribution of breaking strengths in the model is larger (Fig. 6b), fractures can develop at lower strains. This will reduce the overall breaking strength of the aggregate. Since the strength of springs varies significantly, fractures that nucleate may stop propagating. Therefore a range of fractures develops initially with different lengths. More fractures nucleate in an aggregate with a wider distribution of breaking strengths and the devel- oping spacing of fractures will not be as distinct as in the example with the narrow distribution. The material has a bulk response to deformation that is less brittle.

The effect of the mean elastic constant (Young's modulus) of the aggregate on pattern formation and rheological behaviour is shown in Figure 7. The mean Young's modulus of the aggregate increases from Figure 7a to e. Figure 7a to e shows an aggregate with a mean Young's modulus of 1, 2.5, 5, 7.5, and 12.5GPa. All simulations were performed

(a) , (

'. 2 " " '

(b) Cc)

it

! ! 0 ~

i _,oi / �9 , I0 . . . . . . . . . . . . . . 0.01 . . . . . . 0,0~ E

";'F"" I

-~o I ~ - mode,, I~ 0 0,Ol 0.02

Fig. 7. (a) to (e) show simulations under the same pure shear conditions with increasing mean elastic constant. A lower elastic constant produces stronger localization and a wider spacing of mode I fractures. If the elastic constant is higher, mode II fractures dominate. (f) Stress-strain relation for (a). Mode I failure shows a sawtooth curve typical for fast propagation of large mode I fractures. (g) Stress-strain relation for (e). Mode I failure is continuous with no sudden drop of the tensile stress (02 curve). Mode II failure results in a drop of the differential stress.

18 D. KOEHN ET AL.

using pure shear deformation and the snapshots presented are taken at similar amounts of finite strain. The breaking strength of springs in all simulations has the same distribution and mean value. Consequently failure will occur at lower strains in an aggregate that has a higher elastic constant. Therefore the number of fractures increases from Figure 7a to e. In addition, the compressive stresses are not high enough to induce shear failure in Figure 7a and only to a minor extent in Figure 7b and c, whereas shear fractures dominate in Figure 7e. An aggregate with a lower elastic constant localizes exten- sional strain in tensile fractures, which open and form large veins in Figure 7a and to a minor extent in Figure 7b and c. Figure 7d shows a combination of extension fractures and shear fractures. The spacing of extension fractures depends on the mean elastic constant of the aggregate. In the simulation in Figure 7a one large vein develops within the model whereas Figure 7b shows two and Figure 7c three large veins. The spacing is reduced with an increase in Young's modulus. A softer material with a lower elastic constant can relax a larger region around an opening fracture and thus produces a larger spacing than a material with a higher Young's modulus. This relation becomes dis- turbed in Figure 7d and e, where shear fractures start to dominate the pattern. Spacing of shear fractures does not show a simple relation with increasing Young's modulus.

Figure 7f shows the stress-strain relationship for the simulation presented in Figure 7a, and Figure 7g the stress-strain curve for the simulation presented in Figure 7e. These curves illustrate that the rheological behaviour of the aggregate also changes significantly with increasing elastic modulus. Failure due to the development of extension fractures shows significant differences in the stress-strain

relationship of the two examples and failure due to the development of shear fractures only occurs at the finite strain shown in the example of Figure 7g. Figure 7f shows a sudden drop of the tensile stress (0"2) at the beginning of exten- sion fracture development. This indicates that the fractures develop relatively fast and grow large enough to invoke failure of the whole aggregate and reduce the tensile stress almost completely. The overall failure also reduces the differential stress. Figure 7g shows the aggregate with the highest Young's modulus in the pre- sented sequence. It experiences a more continu- ous development of extension fractures that grow progressively while the aggregate reaches higher extensional strains. Therefore the tensile stress is released gradually and the differential stress continues to increase after extension fracture development.

Increase in area

Figure 8 shows three simulations with an area increase of the deformation box. The simulations have horizontal and vertical tensile boundary conditions where the horizontal component is larger in Figure 8a and b. Figure 8a shows a simulation with a large differential stress where the horizontal stress component is three times as large as the vertical component. Figure 8b shows a simulation where the vertical stress component is about 70% of the horizontal stress component. Both extension components in the vertical and horizontal direction are identi- cal in the simulation shown in Figure 8c.

The fracture pattern in Figure 8a represents the dominant horizontal extension. A small number of horizontal fractures develop due to the vertical extension component. Figure 8b shows a differ- ent pattern where the dominance of the horizon- tal extension component can still be seen but the

(a)

,.) r

't' q

.r l

,f ( .

"1 (

>i

(b) .4

i ,-r.

'~ "~ ,'3-

. , , J "], r , ," N .tl

-i s !_~

(c)

�9 , ~ ~ . . '

" ' t "

", J

~ -

. . . . . . . . . ]

Fig. 8. Three simulations with two extension directions and an increase in modelling area�9 (a) Dominant horizontal extension produces well-oriented mode I fractures. (b) Both extension components become more similar so that fractures start to curve�9 Orientation of fractures becomes more diffuse than in (a). (c) Horizontal and vertical extension components are the same, no relation of the orientation of fractures and the model boundaries can be observed. Fractures form polygons similar to mud cracks.

FRACTURE AND VEIN PATTERNS 19

vertical extension component also influences fracture development. Fractures start to curve and lose a dominant orientation with respect to a principal stress direction because the stresses are similar. Fractures still produce a distinct spacing. In Figure 8c the stress field has no domi- nant extension direction and the developing frac- ture pattern shows no preferred orientation with respect to the simulation boundaries. Fracture development and orientation is controlled by the initial noise in the system. Once the first frac- tures develop they influence the stress field in the aggregate, which leads to a polygonization with a distinct spacing similar to the structures found in mud cracks in shrinking sediments (Suppe 1985).

Gravitational loading and pure shear deformation

In a normal tectonic environment one would expect that sedimentary sequences are loaded during their deposition and then experience

tectonic strain afterwards (Price & Cosgrove 1990). In order to investigate the effects of initial gravitational loading on fracture and vein patterns several simulations were performed with complex deformation histories including different tectonic settings�9

In the first simulation with a two-stage defor- mation history we load the sample uniaxially and then apply a pure shear boundary condition. The sample is loaded vertically with fixed side- walls. The vertical stress is proportional to the deformation steps and the horizontal stress is compressive due to Poisson effects. After a given amount of vertical strain we apply a pure shear deformation with vertical constrictional strain and horizontal extensional strain. The developing fracture patterns are shown in Figure 9a in three successive steps. Figure 9b shows the stress-strain diagram for the simu- lation and Figure 9c the corresponding Mohr diagram with four different Mohr circles a to 6 that are indicated in the stress-strain diagram of Figure 9b. The region on the left-hand side

(b) ,ool - t 0 0

0 o

8

0.005 0.01

(c)

(+)

' t

(+)

T

~5

Fig. 9. Complex deformation history involving a compaction event followed by pure shear deformation. (a) Three successive fracture patterns during the simulation. Conjugate mode II fractures dominate the pattern. Only at the latest stage do small mode I fractures develop in the model in order to accommodate local tensile strain. (b) Stress-strain relation of the simulation. Both principal stresses are compressive during the compaction (regime L). Once tectonic deformation starts (regime T) the horizontal stress is relaxed but the differential stress increases. (c) Mohr circle diagrams for (b). The diagram on the left-hand side marks three successive stages during the compaction process. The diagram on the right-hand side shows the successive growth of the Mohr circle during tectonic deformation and the following failure that produces the dominant conjugate mode lI fractures. Note that the x-axis is negative towards the right-hand side since compressive stresses are negative in this paper.

20 D. KOEHN ET AL.

of the vertical line in Figure 9b indicates the gravitational loading regime (marked L for loading). The two principal stresses are compres- sive and the mean stress and differential stress increase successively. The development of the Mohr circle during initial loading is indicated in Figure 9c on the left-hand side where c~ marks the initial starting point and/3 and y the successive growth and movement of the Mohr circle into the compressive regime. After gravita- tional loading pure shear deformation shows an increase in compressive stress and decrease in tensile stress (Figure 9b in tectonic regime marked T). The mean stress stays constant and the differential stress increases with a steeper slope than in the loading regime. At point 6 in Figure 9b failure is initiated so that in Figure 9c the Mohr circle cuts the failure envel- ope. Since differential stress and mean stress are high the developing fractures are shear fractures with typical conjugate sets, which can be seen in Figure 9a. The Mohr circle will not reach the tensile part of the failure curve without crossing the failure curve in the compressive regime if pure shear deformation is applied after the uniax- ial loading. The developing pattern will mainly consist of conjugate shear fractures. Small exten- sion fractures will only grow at a larger amount of deformation in order to accommodate local tensile strains as can be seen in Figure 9a. However, if fluid pressure is involved, the Mohr circle may cross the failure curve in the tensile regime during deformation (see discussion).

Gravitational loading and area increase

In this section we take a look at fracture patterns that develop during different kinds of initial loading followed by a tectonic deformation that is characterized by a large area increase so that both major components of the strain in the mod- elled plane are extensional. The developing frac- ture patterns and corresponding stress-strain curves are shown in Figure 10a to d. Figure 10a shows a simulation with a small amount of initial loading. Tectonic deformation is dominant with a large horizontal extension component. During extension the compressive stresses are relaxed and the horizontal stress component becomes tensile, which results in the develop- ment of vertical extension fractures that are opening. Failure of the whole aggregate is fast, which is illustrated by the sawtooth stress curves in Figure 10a. The material behaves in a brittle manner and shows a strong localization of strain in two to three tensile fractures that are opening and that show a distinct spacing.

Following failure the aggregate reaches a quasi steady-state plastic behaviour where stresses remain at relatively constant values while the strain increases. Small fluctuations in the tensile stress represent growth of secondary extension fractures to accommodate strain.

Figure 10b shows a simulation where gravita- tional loading is dominating. Failure of the aggregate starts during initial loading when a peak differential stress is reached. During this stage conjugate sets of shear fractures develop. Successive tectonic deformation reduces all stress components and leads to a local growth of small extension fractures. The resulting frac- ture pattern is very similar to that of Figure 9, where a smaller amount of initial loading was followed by a pure shear deformation. Both simulations show a dominant primary growth of conjugate shear fractures and a successive development of secondary small-scale extension fractures.

An intermediate pattern between the simu- lations shown in Figure 10a and b is shown in Figure 10c. Here the aggregate is initially loaded by an intermediate amount so that initial loading itself does not induce failure. The aggre- gate is then extended in the horizontal and verti- cal direction where the horizontal extension component is dominant. During failure of the aggregate, conjugate shear fractures develop with a smaller angle towards the principal compressive stress direction than in Figure 10b. The fractures in Figure 10c are of the hybrid extension/shear type of Price & Cosgrove (1990), an intermediate type of fracture. In the simulation shown in Figure 10c these conjugate fractures accommodate successive amounts of extensional strain and are opening.

Figure 10d shows a different type of gravita- tional loading where the aggregate is initially loaded by the same horizontal and vertical amount. Therefore the differential stress stays at zero and the two principal stress components as well as the mean stress increase by the same amount. The following tectonic deformation has a horizontal extension component. During extension all stresses are relaxed. The differential stress starts to increase as a result of the large horizontal extension component. This resulted in the growth of conjugate hybrid extension/ shear fractures similar to those shown in the simulation of Figure 10c.

In summary gravitational loading can lead to the development of primary conjugate sets of hybrid extension or shear fractures that will accommodate extensional strain during succes- sive deformation. It can be seen therefore that fractures and veins in a rock have a record of

FRACTURE AND VEIN PATTERNS 21

(a) 'LJ; ............. ,.i ' ; (b) ,. '...

} , ,

0 (MPa L

zc

/ o~

-20 0

r

AO

O

o~

0.02 0.04 13

100 o ( o

o21 o ~

i

lOO

0 0.01 E

-20 01 -40

-40

0 0.01 0.0Z s 0 O,OOS s

Fig. 10. Four different simulations and the corresponding stress-strain curves for deformation histories involving different kinds of compaction that are followed by tectonic deformation that involves extension in the modelling plane. (a) Small amount of compaction results in a situation where mode I fractures are dominant. (b) Large amount of compaction results in the development of conjugate mode II fractures during compaction. The following tectonic extension produces local mode I fractures but strain is mainly accommodated by the already existing fracture network. (c) A simulation with intermediate compaction followed by tectonic extension results in the growth of hybrid extension/shear fractures that are opening. (d) Compaction that is hydrostatic can also lead to hybrid extension/shear fractures if the following tectonic deformation is strongly non-hydrostatic so that extension is dominating in the horizontal direction.

22 D. KOEHN ETAL.

the gravitational loading of the rock prior to tectonic deformation.

Discussion

We compare the results of the numerical study with the vein sets shown in Figure 2. The vein in Figure 2a may have started as a hybrid exten- sion fracture and may reflect a stress state that would produce that type of fracture. This stress state would then be represented by the whole history of the vein development. However, our simulations show that quite a number of different stress states may develop during a deformation history. Therefore this vein could also form as a shear fracture (or hybrid shear depending on its orientation relative to the principal stress axis) initially and then open afterwards due to an extension component during deformation. Such a setting could also produce vein patterns similar to the ones shown in Figure 2b where conjugate shear (or hybrid shear) fractures may form first and then later extension takes over and the initial fractures are opening as veins. However, further study on the internal geome- tries of such vein sets is needed in order to prove that the proposed scenario can develop in nature.

We use hyperbolic Mohr circle diagrams to illustrate the stress states and failure in our simulations. There is still much debate on whether a hyperbolic Mohr circle can be used for intermediate failure types between mode I and mode II, namely hybrid extension/shear fractures. Engelder (1999) illustrates that these intermediate structures cannot be explained as propagating cracks by linear elastic fracture mechanics theory. He rather concludes that they may form by an out-of-plane propagation of mode I fractures that are subject to a shear stress. In our simulation a number of scenarios produce patterns that are dominated either by mode I or mode II fractures. In intermediate set- tings quite often a combination of mode I and mode II fractures develop where both failure modes still show distinct sets of fractures. Only in a narrow range do we find fractures that have an intermediate orientation (between mode I and II), which fall into the category of hybrid extension/shear fractures (Fig. 10c). These may well be combinations of mode I and mode II fractures as proposed by Engelder (1999).

It is interesting to see that in a number of simu- lations quasi steady-state behaviours are reached (Figs 5a and 10a, c). The fracture systems that develop can accommodate additional strain with only minor increase in stress, which is

probably due to additional growth of small fractures or friction along shear fractures. Prior to this quasi steady-state behaviour, stresses drop significantly when the whole systems fails. The remaining tensile stress component is very small since most tensile stresses can be accom- modated by opening fractures or veins. Compres- sive stresses are higher and can probably be related to friction along shear fractures as long as one component of the stress tensor in the mod- elling plane is compressive. As long as the system does not heal after failure stresses will probably remain low. If this steady state is also reached in natural rocks, it has a strong influence on the strength of brittle faults and earthquake behaviour (Scholz 2002). Once a rock fails by fracturing it may deform in a plastic way without significant increase in stress (Paterson 1978; Zhang et al. 1990; Karato 1995). Once a fault forms and develops a cataclasite it may behave as a plastic material with no major increase in stress if fractures are not healing. Therefore no successive earthquakes will be expected along such a fault. However, if veins grow and fill the fractures, the cataclasite may heal and retain a certain strength again. Then it may fracture by catastrophic failure at high stresses (Scholz 2002).

What we have not yet taken into account is the effect of fluid pressure on the developing fracture patterns. This is certainly an important com- ponent that we will include in the future. A high fluid pressure can shift the Mohr circle towards the tensile regime so that mode I fracture may develop during the loading history instead of conjugate shear fractures (Price & Cosgrove 1990; FlekkCy et al. 2002). However, conjugate shear fractures may also develop in this case if the differential stress is already high due to a non-hydrostatic initial loading. Fluid pressure and developing fractures will influence each other depending on how effectively fractures can drain an existing fluid pressure. If the tec- tonic stress is non-hydrostatic or if fluid pressure gradients exist along anisotropies, fluid pressure will provoke failure (Bercovici et al. 2001). Once fractures develop, the fluid pressure may drop because the permeability of the rock increases. Whether or not the fluid pressure influ- enced fracture and vein development in the natural samples that were presented in this paper is not clear. Bedding parallel veins in the area may have formed during high fluid pressures because they are located along the bedding planes between porous sandstones and less porous mudstones. Fluid pressure gradients can be expected along these anisotropies, which may invoke failure. The veins that are presented

FRACTURE AND VEIN PATTERNS 23

in this paper are however located within the more porous sandstones and are probably more associated with tensile strain than a higher fluid pressure. However, additional research is needed in order to fully understand the effect of fluid pressure on the development of these systems.

In the following paragraphs we will discuss some additional boundary effects of the model that we used. The box boundaries have an influ- ence on the development of shear fractures. Box boundaries reflect shear fractures because the boundaries behave like rigid pistons and bound- ary particles are fixed perpendicular to the walls. They cannot accommodate a shear displa- cement of the boundary so that shear fractures that hit the walls are repelled and a new shear fracture grows back into the model. This effect will influence the density of shear fractures in the model and also their spacing. In some cases shear fractures will dominate near to the com- pressive walls (upper and lower wall) whereas extension fractures dominate in the centre of the model (Fig. 7). However, this effect is not always observed (Fig. 10c).

An additional problem arises because the underlying lattice is triangular. Therefore frac- turing is partly anisotropic, which is especially important in the case of shear fractures where slip should take place. The anisotropies of the lattice can be overcome by adding distributions to the elastic constants and spring breaking strengths that are larger than the lattice anisotro- pies (Malthe-SCrenssen et al. 1998b). Mode I fractures show no lattice-preferred orientation in our model, which can be best observed in Figure 8c where stresses are hydrostatic and no lattice directions nor any effects of the model boundaries are observed. Shear fractures may however still be influenced by the lattice (Fig. 10b) but can also develop in non-lattice directions (Fig. 10c). One has to note that the direction of the shear fractures in Figure 10b is more or less a lattice direction, but also the direc- tion where the fractures would develop anyhow.

The threshold of relaxation in the model can also change the developing pattern. This is recorded in detail in Malthe-SCrenssen et al. (1998b). Since we are moving all particles assuming homogeneous strain in the model before relaxation starts and since our relaxation threshold is very small relative to the applied strain, it has no major influence on the develop- ing pattern. However, if a significantly higher relaxation threshold is used, different spacing of mode I fractures will result since the system will be progressively slower and relaxation will be more localized.

Conclusions

Modelling of progressive fracture development in aggregates shows that different type of frac- tures can grow during one tectonic deformation event. Pure shear deformation produces primary extension fractures that are followed by a sec- ondary set of conjugate shear fractures. These patterns change depending on the properties of the aggregate, namely its mean elastic constant and the initial noise in the system. Pure extension of the modelling area leads to one dominant extension fracture set or to a polygonal set of fractures if the principal stresses are equal. A more complex deformation history that includes gravitational loading as well as tectonic strain can lead to the development of conjugate shear fracture sets that can accommodate extensional strain and thus form veins during later stages of deformation. These conjugate sets of veins may be followed by secondary extension fractures. These complex veins may record gravitational loading preceding the tectonic deformation. Their orientation can be used to find the orien- tation of the non-hydrostatic stress field that existed during gravitational loading or tectonic deformation.

We thank T. Engelder and J. Cosgrove for constructive reviews. We also thank A. Malthe-SCrenssen for his help with the discrete element code. JA acknowledges funding by the DFG-Graduiertenkolleg 'Composition and Evolution of Crust and Mantle'.

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