Influence of fracture geometry on shear behavior

$: Influence of fracture geometry on shear behavior$
In¯uence of fracture geometry on shear behavior

S. Gentiera,*, J. Rissb, G. Archambault c, R. Flamandc, D. Hopkinsd

aBRGM, 3 avenue Claude Guillemin, BP 6009, 45060 OrleÂans ceÂdex 2, FrancebCDGA UniversiteÂ Bordeaux I, Avenue des faculteÂs, 33405 Talence ceÂdex, France

cCERM, UniversiteÂ du QueÂbec aÁ Chicoutimi, 555 boulevard de l'UniversiteÂ, Chicoutimi, QueÂbec, Canada, G7H2B1dLaurence Berkeley National Laboratory, One Cyclotron road, MS 46A-1123, Berkeley, CA 94720, USA

Accepted 7 October 1999

Abstract

Laboratory research during the past 10 years has explained many critical links between the geometrical characteristics offractures and their hydraulic and mechanical behavior. One of the remaining research challenges is to directly link fracture

geometry with shear behavior, including behavior in response to changes in normal stress and shear direction. This paperdescribes results from a series of shear tests performed on identical copies (replicas) of a natural granite fracture. Based on thesetests, we developed a method using image processing techniques to identify and quantify damage that occurs during shearing.We ®nd that there is a strong relationship between the fracture's geometry and its mechanical behavior under shear stress and

the resulting damage. Using a three-dimensional geostatistical model of the fracture surfaces, we analyze the dependence of thesize and location of damage zones on the local geometry and propose an algorithm for predicting areas that are most likely tobe damaged during shearing in a given direction. 7 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Fracture geometry; Fracture topography; Geostatistics; Image processing; Jointed rock masses; Kriging; Directional gradient; Rock

joint; Shear behavior; Shear test; Surface roughness

1. Introduction

Prediction of the hydromechanical behavior of frac-tured rock masses at the ®eld scale is important for avariety of activities, including oil and geothermal pro-duction, underground excavation, and mine design.Understanding changes in the behavior of fracturedrock resulting from changes in stress improves ourability to assess the stability of excavations, and topredict changes in ¯uid ¯ow, and improves our under-standing of induced seismicity.

Laboratory research during the past 10 years hasmade many critical links between the geometricalcharacteristics of fractures and their hydromechanicalbehavior under normal stress [1±3]. Understanding the

relationship between ¯uid ¯ow and geometry for frac-

tures in shear is more complicated than for fractures

under only normal stress because of the need to

account for the creation of damage zones. Linking

fracture geometry to ¯ow and changes in ¯ow that

occur with changes in shear stress and direction is,

however, beyond the scope of this paper. Before we

can understand the hydraulic behavior of fractures in

shear, we must ®rst understand the mechanics of

shear, in particular, mechanical behavior in response

to changes in normal stress and shear direction. An

essential step in gaining this understanding is making

the link between fracture geometry and shear behavior.

In this paper, we take a ®rst step toward this work by

describing results from a series of shear tests per-

formed on replicas of a natural fracture. Because the

replicas were identical, we were able to study the

mechanical response of samples with the same fracture

geometry to a variety of shear conditions (three values

International Journal of Rock Mechanics and Mining Sciences 37 (2000) 161±174

1365-1609/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.

PII: S1365-1609(99 )00096 -9

www.elsevier.com/locate/ijrmms

* Corresponding author. Tel.: +33-2-38-64-38-88; fax: +33-2-38-

64-33-61.

E-mail address: [email protected] (S. Gentier).

of normal stress and four shear directions). We showthat all mechanical parameters are dependent on sheardirection.

Analysis of our mechanical results, in conjunctionwith images and pro®les of sheared fracture surfaces,has helped clarify the relationship between the frac-ture's geometry and shear behavior. Before the sheartests are performed, pro®les of the fracture surfacesare measured, and maps of the topography of eachsurface are created from the pro®les using geostatisti-cal methods. After shearing, images of the fracture sur-faces are analyzed to identify damaged areas. Maps ofthe damage zones are superimposed onto the geostatis-tically constructed maps of the fracture surfaces tocharacterize the damage with respect to shear direc-tion, normal stress, and the local geometry of the sur-faces. Comparison of the pro®les of fracture surfacesobtained before and after shearing, together with theanalysis of the images of sheared surfaces, helps usquantify the damage that occurs during shearing anddevelop a method to predict its magnitude and lo-cation based on the initial geometry of the fracturesurfaces. To link fracture geometry directly to shearbehavior, we propose an algorithm that predicts the lo-cations on the surfaces of a fracture that are likely tobe damaged during shearing.

2. Experimental procedures

The work was carried out on identical replicas of agranite sample so that shear tests could be performedon fractures with the same surface topography. Thesample was cored across a natural joint of GueÂ retgranite from France [4]. The fracture is located half-way up the cylindrical core sample and perpendicularto its axis; the diameter of the core sample is 90 mm.

2.1. Sample preparation and characteristics

Molds of both walls of the fracture were made bycasting with a silicon elastomer. These molds werethen used to create replicas from non-shrinking cementmortar. The fracture's upper walls were left gray, andthe lower walls were stained pink in order to highlightmaterial exchange during shearing. Mean mechanicalproperties of the mortar are shown in Table 1 [5].

Table

2

Combinations{y,ÿ

sN,Du}usedforsheartests

7MPa

14MPa

21MPa

ÿ308

±0.7

1.4

±5.1

±0.8

±2.5

4.9

±0.8

±2.2

5.0

08

0.3

0.5

1.0

2.0

5.0

0.3

0.5

1.0

2.0

5.0

0.3

0.5

1.0

2.0

5.0

608

±0.7

±2.1

5.0

±±

1.0

2.5

5.0

±±

1.1

2.6

5.0

908

±±

1.0

1.4

5.0

±±

1.2

2.2

5.1

±0.8

±2.8

5.0

Table 1

Mechanical characteristics of the mortar used to make fracture repli-

cas: compressive strength (C0), tensile strength (T0), friction angle

(Fm), and Young's modulus (E)

C0 T0 Fm E

75 MPa 6.6 MPa 358 30 800 MPa

S. Gentier et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 161±174162

Fig. 1. Schematic representation of the series of replicas (a) and de®nition of shear directions shown on the lower wall of the fracture (b).

Fig. 2. Direct-shear stress test results: shear stress vs shear displacement and shear stress vs dilatancy in relation to shear direction, y (see Fig. 1).

S. Gentier et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 161±174 163

2.2. Shear test device

The experimental device used is a classical shear ma-chine that is servocontrolled in force or displacement.The shear machine consists of two parallelepiped half-boxes: one ®xed and one that moves. The lower, ®xedhalf-box, was bound to the frame and rested againstthe back of a horizontal jack. The upper, moving half-box was pulled by the rod of the horizontal jack. Thisdevice had three degrees of freedom:

1. a translation along x (shearing direction);2. a translation along z (direction perpendicular to the

shearing plane); and3. a rotation around y (direction perpendicular to the

shear direction in the shear plane).

The normal stress, sN, was applied by a second, verti-cal jack attached to a moving frame, so the jack couldfollow the upper moving half-box. An inclinometertracked movement of the frame continuously duringthe shear tests to provide a record of any tilting. Twodigital servo-regulators controlled the oil pressure inthe two hydraulic jacks. Normal displacement, Dv,shear displacement, Du, inclination of the upper pla-teau of the moving half-box, and pressure in the hori-zontal and vertical jacks were recorded during sheartests.

2.3. Shear test program

Shear tests were performed under applied normalstresses, sN, of 7, 14, and 21 MPa. For each normalstress, shearing was permitted, in separate tests withdi�erent samples, to continue to several di�erent levelsof displacement as indicated in Table 2 [5]. The shearvelocity was 0.5 mm/min. Each test was repeated forfour di�erent directions (08, ÿ308, 608, 908) denotedby y (Fig. 1). For each change of variable, a singlefracture replica was used. Thus, each mortar copy wasused only once, for a single combination of stress, dis-placement, and shear direction. Table 2 shows thecombinations {y, sN, Du } that were used for sheartests.

3. Mechanical results

We can divide the shearing process into ®ve phases[6,7]:

1. The elastic phase, with reduction in aperture on thejoint plane as a result of sN. Contact area and shearsti�ness increase in this phase, and there is a re-duction in the volume of the void space.

2. The non-linear dilatancy phase (up to peak shearstress), dominated by dilatancy and local redistribu-T

able

3

Dependence

ofmechanicalcharacteristics

onsheardirection

t p(M

Pa)

Ks(M

Pa/m

m)

DU

p(m

m)

i p(8)

t r(M

Pa)

s N(M

Pa)

714

21

714

21

714

21

714

21

714

21.0

ÿ308

9.78

17.82

22.86

37.61

60.10

64.77

0.366

0.443

0.633

12.33

11.80

6.92

6.0

10.5

15.7

08

10.25

18.37

24.25

33.76

51.40

68.49

0.423

0.502

0.599

14.22

12.32

10.07

6.3

11.5

17.0

+608

11.41

19.98

25.94

32.84

50.69

59.92

0.536

0.644

0.767

12.50

13.75

10.69

7.4

13.5

19.0

908

10.05

17.99

25.99

18.95

29.70

33.44

0.905

1.000

1.140

12.49

9.57

9.99

7.9

14.6

20.7


tion of stress resulting from deformation and fric-tional slip at individual contacting asperities (iso-lated small asperities may break).

3. The peak shear stress phase, with failure of asperi-ties and a maximum dilatancy rate.

4. The post-peak phase, a progressive softening (un-stable yielding) phase, in which there is progressivedegradation of the fracture surfaces (from micro-fracturing of asperities, crushing, and failure ofasperities, all depending on sN). There is an increasein the contact area between the two surfaces of thefracture with a corresponding decrease in normalstress acting on the contact areas. Dilatancy is stillincreasing but at a lower rate than during phase 3.

5. The residual strength phase (stable sliding) in whichshear and normal stresses are relatively stable, butdegradation on joint surfaces is still occurring.

Although shear behavior is qualitatively similar fordi�erent shear directions, the main characteristics(peak shear stress, tp; residual shear stress, tr; dila-tancy at the peak, ip; shear sti�ness, Ks; horizontal dis-placement at the peak, DUp; etc.) have all been shownto be dependent on the shear direction (Table 3).

Fig. 2 shows key experimental results. The fracture'smechanical response di�ers depending on the sheardirection. The mechanical behavior falls into two cat-egories. The ®rst corresponds to the ÿ308 and 08 direc-tions (parallel to the strike of the fracture), and thesecond to the +608 and +908 directions (parallel tothe dip of the fracture). The fracture's strike and dipare calculated from the mean fracture plane derivedfrom a multivariate linear regression z=f(x,y )

described by Riss et al. [8]. Because results of labora-tory experiments on samples with identical fracturegeometry show that mechanical parameters depend onshear direction, we attempt in the next sections todetermine the role of geometrical characteristics, inparticular, surface roughness, in shear behavior.

4. Damaged areas

Degradation on the surface of the replicas consistedof gouge material: pink material was found on thegray (upper) surfaces of the fractures and gray ma-terial on the pink (lower) surfaces (Fig. 3). However,most gouge material was white as a result of myloniti-zation (crushing) during shearing. The greater the nor-mal stress and displacement, the more white materialresulted. Some gouge material from both joint faceswas dislodged without being stuck to the oppositeface. Gouge material was not uniformly distributed.

Because the gouge material was mostly white, ablack and white CCD (charged±coupled-device) cam-era was used to make images of the upper and lowersurfaces of the fracture after shear testing. All theimages were taken under identical conditions with eachfracture replica placed in the same position relative tothe camera and light sources. The images were digi-tized using 512 horizontal pixels, 512 vertical pixels,and 256 gray levels (Figs. 4, 5, 6a). Image analysis wasthen used to quantify damage on the joint surfaces,using the segmentation method described in Riss et al.[9,10]. Results are presented in binary images (black

Fig. 3. Color images of the sheared surfaces (sN=21 MPa, Du=5 mm, y=08).


and white) that show the size, shape, and spatial distri-bution of the areas damaged during shearing (Figs. 6band 7). We conclude that the size, shape, and spatialdistribution of damaged areas depend on the sheardirection and the degree of stress and horizontal dis-placement; as would be expected, damage increases asstress and displacement increase.

It does not appear that any major damage occursprior to the peak shear stress. Damage occurred pri-marily during the softening and residual phases; how-ever, although visible damage was absent during the®rst two stages, the locations of contacting asperitiesduring these phases correlated closely to the damagedareas that appeared later. For a speci®ed surface, thesedamage zones expand in a direction opposite to thedirection of displacement of the surface. Damage startsaround the areas of the fracture with the steepestslopes. The shape of the damage zones depends on thelocal geometry of the fracture surface, including thesize and shape of asperities. The extension of thedamage zones in the shear direction depends on thefracture's horizontal displacement. In any damagedarea, asperities can be located on top of each other,making contact at their tips, or in any kind of side-by-side con®guration. Because the nature of the asperities

is an important factor for modeling shear behavior, weshow in the following section how to get accurate geo-metrical information about damage zones and howthey may be predicted.

5. Three-dimensional modeling of the fracture surface

The spatial features of a fracture can be studiedusing geostatistical methods [11]. The overall objectiveof a geostatistical approach is to characterize thespatial variations of the fracture topography and to in-corporate them into modeling the fracture, using tech-niques including kriging. In statistical terms, kriginginvolves using an unbiased linear estimator with aminimal quadratic error to estimate surface elevation(z ).

A set of pro®les was recorded in each of thefour shear directions, resulting in 7563 (lower sur-face) and 7556 (upper surface) coordinates {x, y,z }. After the experimental variograms of elevationsz and ®rst derivatives z ' were calculated, a theoreti-cal variogram was computed and ®t to the exper-imental data. A good ®t to the experimental datawas achieved using a theoretical variogram obtained

Fig. 4. Grey-level images showing the evolution of damaged areas with increasing horizontal displacement (constant normal stress sN=21 MPa,

y=08).

Fig. 5. Grey-level images showing the evolution of the damaged areas with increasing normal stress (constant horizontal displacement Du =

5 mm, y=908).


by superposition of three basic models (Gaussian,spherical, and quadratic rational; see Riss et al. [12]and Gentier et al. [13]). Although kriging based onthis theoretical variogram smoothes local variations,it provides a good representation of surface topo-graphy (Fig. 8). Furthermore, because {x, y } dataaround the boundaries of the fracture's damagedareas were computed from image analysis of thesurfaces, {z } values can be calculated from the pre-vious three-dimensional geostatistical reconstructionin such a way that the damaged areas can beimaged in three dimensions (Fig. 9). Additional in-formation (e.g., roughness, local dipping, and localmaximum elevation) is available from this three-dimensional representation of the damaged areas. Aspeci®c example (sN=21 MPa, y=+908, Du=5 mm,Dv = 0.68 mm) illustrates general results; Fig. 10a,bshows the relationships among the gradient (i.e., ®rstderivatives z ' in the vertical plane of the pro®le) of theactual pro®les of the upper and lower surfaces [z,before (a) and after (b) shearing], and the luminance(light intensity) of the gray-level image of the fracture

along the lower pro®le. We also added the luminance(light intensity) of the lower pro®le, shown in part inFig. 11, from a color photograph of the fracture takenafter shearing and then scanned. Damaged asperitieswith a luminance higher than the threshold that wasused to create the binary images (see Fig. 10a,b) corre-lated well with positive gradients. In fact, there arethree types of damage (Fig. 11): (1) actual crushedwhite material (easy to see on black and white images)slightly shifted in the direction opposite to shearingwhere gradients are steep; (2) uncrushed material fromthe opposite surface, detected because it retained itsoriginal color, occurring adjacent to areas of maximumgradient; and (3) gouged areas that produce type 2 ma-terial. In other words, we encounter successively: graystuck material, white crushed material, and darkgouged areas along a line parallel but opposite to theshear direction on the lower pink wall (along a linedrawn symmetrically on the gray upper wall, we ®nd apink stuck material, a white crushed material, anddark gray areas).

We conclude that damaged areas predominated

Fig. 6. (a and b) Grey-level (top) and binary images (bottom) of the damaged areas of both upper and lower surfaces of the fracture (sN=21

MPa, Du=5 mm, y=ÿ308).


Fig. 7. Size, shape, and spatial distribution of damaged areas occurring during shear tests in relation to shear direction and normal stress (Du=5 mm).


Fig. 8. Three-dimensional reconstruction of the fracture surfaces (Kriging method).

Fig. 9. Damaged zones superimposed on fracture topography (sN=21 MPa, Du= 5 mm). On the left, shearing directions (ÿ308, 08) are nearly

parallel to the strike direction (N±S) of the fracture; on the right the shearing directions (608, 908) are nearly parallel to the dip direction of the

fracture (E±W).


where the local dip direction was close to the sheardirection, and also that areas with higher dip valueshad the highest probability of damage. Assuming thisconclusion applies to any stress (sN) and shear direc-tion (y ), it is possible to predict, using three-dimen-sional representation of the fracture geometry, whichasperities on a fracture's surface will likely be damagedduring shearing.

6. Prediction of the mechanical damage

Mechanical damage to a fracture as a result of shearstress can be predicted using image analysis, which isvery e�ective for assessing the roughness of non-planarsurfaces such as fracture walls. We created a gray-levelimage of the lower wall of the fracture (Fig. 12),assuming that the gray levels f(z ) are proportional tothe elevations (z ) of the fracture surface.

Because the slope of the asperities correlates withthe likelihood that they will be damaged during shear-ing, we analyzed the gray-level image using the direc-tional gradient, which is an image-processing operatorthat is used to calculate the slope of the surface in aspeci®ed direction. The directional gradient gy (f) of afunction f in the direction y is de®ned by the di�erencebetween the `thickening' and the `thinning' of the func-tion f (thickening and thinning are classical morpho-logical transformations operating on the gray levels ofthe images [15]):

gy�F� � �F� Ty� ÿ �F0Ty�

where T1 and T2 form the two-phase structuring el-ement Ty � �T1, T2�y: The procedure for computingthe directional gradient is given in [14]. The result ofthe operation is to change the gray level of the currentpixel xi depending on the di�erence in gray levels at

Fig. 10. (a and b) Relationships between damaged areas and topography.


adjacent pixels xi ÿ 1 and xi + 1 in the y direction; thenew gray level at xi is proportional to the di�erencebetween the gray levels at adjacent pixels for di�er-ences greater than zero (indicating positive slope), orblack for di�erences less than or equal to zero (indicat-ing negative or zero slope). Fig. 13 (2nd row) showsbinary images with the locations of the main direc-tional gradients that were computed for each of theshearing directions. The locations of the maximumgradients correspond well to the locations of damagezones for a normal stress of 21 MPa for all sheardirections. It must be noted that the damaged areasoccurring under normal stresses of 7 MPa and 14 MPaare subsets of the damage zones that result from a nor-

mal stress of 21 MPa. When the shear direction is par-

allel to the dip direction, the damaged areas are

strongly correlated to the directional gradient. The

same correlation is also apparent when the shear direc-

tion is parallel to strike but is less obvious because

topographical features in this direction are dispersed

and not contiguous. Therefore, we conclude that the

directional gradient is a good predictor of the areas of

the fracture surface that will be damaged during shear-

ing.

The areas of maximum directional gradient occur

either on the sides of asperities or on the sides of crest

lines; these locations can be de®ned readily, but they

do not indicate the full extent of future damaged

areas. Assuming that the volume of material that will

be damaged during shearing is located all around

points of maximum gradient but preferentially in the

shearing direction, we propose that damaged areas can

be inferred by image analysis methods [15] using the

following steps (Fig. 14):

1. Computing the gray-level gradient for a given direc-

tion, i.e., directional gradient (Fig. 14b and c).

2. `Thresholding' the image obtained from Step 1 to

get a binary image that more clearly shows the

areas of highest directional gradient (Fig. 14d).

3. Filtering the binary image using an image-analysis

operator (structural element with opening of size 1)

to remove small isolated features, followed by

reconstruction (Fig. 14e). The ®ltering procedure

removes all features smaller than the de®ned struc-

tural element. The purpose of reconstruction is to

Fig. 11. The three types of damage during shearing.

Fig. 12. Grey-level image of the topography of the lower wall of the

fracture. Grey levels F(z ) are proportional to the elevations (z ) of

the fracture surface. Grey levels are computed using the equation

F(z )=aZ+b. Z values are the elevations calculated at the grid nodes

by the geostatistical simulation. F(z )$ [5250], pixel size is 0.5 mm2,

and each gray level represents 33 mm.


recover the deleted areas that are not isolated butare instead part of larger features.

4. `Dilating' this binary image of maximum gradientto expand the predicted damage zones using a direc-tional structuring element with a size correlated tothe horizontal displacement (Fig. 14f). Steps 1±3identify the locations of steepest gradient; the pur-pose of dilation is to expand these areas in theshear direction to account for damage that occurswith increasing shear displacement.

5. Determining the symmetrical di�erence between theimages of the gradient and dilation of the gradient(Fig. 14g). The symmetrical di�erence is de®ned interms of basic set theory: for two sets X and Y, thesymmetrical di�erence X\Y=(X [ Y)ÿ(X \ Y). Forthe results presented here, X corresponds to thehigh-gradient areas determined in Steps 1±3, and Ycorresponds to the area after the dilation of X (Step4). The symmetrical di�erence X\Y corresponds tothe area added during directional dilation. Although

one might expect damage to initiate in the highest

gradient areas identi®ed in the ®rst three steps, the

best ®t with experimental data is obtained when

these areas are subtracted from the dilated area by

taking the symmetrical di�erence de®ned above.

Fig. 14h and i illustrate the results of applying the al-

gorithm to the experimental data for shear direction

y=908. The predicted damage areas (Fig. 14h) corre-

late closely with the actual damage (Fig. 14g) deli-

neated by white gouge material (damage from

crushing). We conclude that our algorithm e�ectively

predicts damage to fracture surfaces that occurs during

shearing and is the ®rst method to do so. The next

step in this research is to de®ne an algorithm for

damage prediction that is based on deterministic cri-

teria; for example, setting the threshold used in Step 2

according to the normal stress and the size of dilation

(Step 4) according to the horizontal displacement

(Du ).

Fig. 13. Directional gradient for the lower surface of the fracture (2nd row). Contoured areas show the damaged areas (white pixels) that

occurred during shearing with sN=21 MPa and Du=5 mm (1st row) and with sN=7 MPa and Du=5 mm (3rd row).


7. Conclusions

Laboratory experiments performed on replicas of anatural granite fracture show that the mechanicalbehavior of fractures under shear stress is strongly re-

lated to the geometry of the fracture surfaces. Mechan-ical parameters measured during experiments varieddepending on the direction of shear. Although themechanical behavior was qualitatively similar fordi�erent shear directions, primary parameters all

Fig. 14. (a±i) Application of successive steps of the algorithm for prediction of damaged areas and predicted damaged areas (for shear direction

908, sN=21 MPa and Du=5 mm).


depended on the shear direction; these parametersinclude peak shear stress, residual shear stress, dila-tancy and displacement at peak shear stress, and shearsti�ness.

Analysis of images of damage zones on fracture sur-faces after shearing suggests that there is little damageduring the ®rst two phases of shearing, prior to thepeak shear stress. Damage is initiated in areas of thefracture with the steepest slopes. The damage zonesexpand during the post-peak softening and residualphases of shear. The size, shape, and spatial distri-bution of the damaged areas depend on the sheardirection as well as on the level of stress and shear dis-placement. The extension of the damage zones in theshear direction depends on the amount of horizontaldisplacement during shearing.

Fracture surfaces reconstructed from pro®les usinggeostatistical methods were compared to damage zonescreated during shearing characterized using image pro-cessing techniques. This comparison showed thatdamage is most likely to occur in areas where the localdip direction is close to the shear direction and inareas with the steepest gradients.

The experimental results and results from the com-parison between fracture surface reconstructions anddamage zone images were used to construct an image-analysis-based algorithm that predicts where damagewill occur. Directional gradients are calculated fromthe images of the surfaces. It is assumed that damagewill occur around areas of maximum gradient but pre-ferentially in the shear direction. The areas of maxi-mum gradient occur either on the sides of asperities oron the sides of crest lines. To achieve a good match tothe damage zones observed on laboratory samples, theareas of maximum gradient predicted to be damagezones must be expanded in the shear direction by anamount related to the shear displacement. The lo-cation, size, and shape of damage zones predicted bythe algorithm agree closely with the damage observedon laboratory specimens.

Results of laboratory experiments, used in conjunc-tion with images of fracture surfaces and modelingthat allows reconstruction of fracture topography fromsurface pro®les, provide a powerful tool for linkingfracture geometry to the mechanical behavior of frac-tures under shear stress. Studies such as this one thatutilize data obtained from a variety of techniques arelikely to be fundamental in formulating a constitutivelaw of shear behavior that incorporates fracture geo-metry.

Acknowledgements

BRGM (France) and the University Bordeaux I

(France) supported the research described in thispaper.

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Influence of fracture geometry on shear behavior

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