Quantitative Parameters for Rock Joint Surface Roughness

Rock Mech. Rock Engng. (2000) 33 (4), 217±242

Rock Mechanics and Rock Engineering : Springer-Verlag 2000 Printed in Austria

Quantitative Parameters for Rock Joint Surface Roughness By

T. Belem, F. Homand-Etienne, and M. Souley Laboratoire Environnement GeÂomeÂcanique et Ouvrages, Ecole Nationale SupeÂrieure de GeÂologie, Vandúuvre-les-Nancy, France

Summary The morphologies of two arti®cial granite joints (sanded and hammered surfaces), one arti®cial regularly undulated joint and one natural schist joint, were studied. The sanded and hammered granite joints underwent 5 cycles of direct shear under 3 normal stress levels ranging between 0.3±4 MPa. The regularly undulated joint underwent 10 cycles of shear under 6 normal stress levels ranging between 0.5±5 MPa and the natural schist replicas underwent a monotonous shear under 5 normal stress levels ranging between 0.4±2.4 MPa. In order to characterize the morphology of the sheared joints, a laser sensor pro®lometer was used to perform surface data measurements prior to and after each shear test. Rather than describing the morphology of the joints from the single pro®les, our characterization is based on a simultaneous analysis of all the surface pro®les. Roughness was viewed as a combination of a primary roughness and a secondary roughness. The surface angularity was quanti®ed by de®ning its three-dimensional mean angle, ys , and the parameter Z2s . The surface anisotropy and the secondary roughness were respectively quanti®ed by the degree of apparent anisotropy, ka , and the surface relative roughness coe½cient, Rs . The surface sinuosity was quanti®ed by the surface tortuosity coe½cient, Ts . Comparison between the means of the classical linear parameters and those proposed shows that linear parameters underestimate the morphological characteristics of the joint surfaces. As a result, the proposed bi-dimensional and tri-dimensional parameters better describe the evolution of the joints initial roughness during the course of shearing.

1. Introduction One of the major problems in the morphological characterization of rock joint surface roughness is related to the choice of adequate parameters. Furthermore, this choice is only possible if we are already able to answer to the following question: can roughness be su½ciently described by only one parameter? One could think, a priori, that a single parameter does not su½ciently characterize the roughness which includes such morphological characteristics as magnitude (surface point elevations), angularity (slopes and angles), undularity (periodicity), anisotropy, and, in a less pronounced way, curvature. In addition, can the morphological characterization of a surface with certain pro®les (with the unknown

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minimal representative number) be considered su½cient? One knows that the notion of roughness implicitly includes that of the surface. This roughness is most often described by some linear parameters calculated from individual pro®les obtained along one-directional parallel lines (x and/or y directions). The parameter describing the entire surface of the fracture can be obtained by calculating the average of the parameters of all the pro®les (pseudobi-dimensional parameter). Here, this procedure was quali®ed as an indirect method (e.g. indirect characterization of the morphology), which was designated by some authors as the detailed study of the roughness (Riss et al., 1995) or the characterization of the local roughness (Wan et al., 1996). The direct characterization of the morphology consists of describing the morphology by a simultaneous analysis of all the surface pro®les of the fracture. This procedure was referred to by Wan et al. (1996) as the characterization of the global roughness. Until today no satisfactory method for the characterization of roughness has been established as a reference. Most of the time, three-dimensional problems are tackled by using two-dimensional approaches, except for a few attempts of 3D modeling (Brown and Scholz, 1985; Yoshioka and Scholz, 1989; Riss and Gentier, 1990). For the purpose of elaborating a constitutive law for sheared rock joints explicitly taking into account the evolution of the initial morphology, it seems necessary to better understand the exact role played by this morphology during the course of shearing. To achieve this objective, we need both sur®cial and 3D parameters whose range of variation allow to take into account the evolution of the state of surfaces in contact during the course of shearing. In this paper, we propose an approach of the surface pro®lometric data analysis for the characterization of the joint surface roughness using the primary and secondary asperities concept (Jing et al., 1992; Kana et al., 1996). The secondary asperities (e.g. second order roughness or sensu stricto roughness) are de®ned by the distribution of surface heights, while the primary asperities (e.g. ®rst order roughness) are de®ned by the global geometry of the surface. At ®rst, secondary roughness is characterized in terms of angularity and sur®cial roughness. Based on the existing linear parameters Z2, RL , and Ps , we de®ne their sur®cial and/ or 3D equivalents Z2s , Rs , and Ts . Indeed, equations of these geometric linear parameters show their sensitivity to the variations of both the amplitude and the periodicity of roughness pro®les. A linear angularity parameter, yp , and its 3D equivalent, ys , are de®ned in order to extend the previous three parameters. In the second phase, primary roughness is characterized in terms of real and apparent structural anisotropy, geometric irregularity and undularity of the surface. The real structural anisotropy is characterized through the geostatistical analysis (variogram and auto-correlation function), while the apparent structural anisotropy is characterized with the aid of the chosen linear parameters in this study (yp , yp‡ , ypÿ , Z2, RL and Ps ). The geometric irregularity is quanti®ed using only the linear angularity parameters (yp‡ and ypÿ ).

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2. Characterization of Secondary Roughness In order to compare the 3D and/or sur®cial parameters to their corresponding linear parameters, we ®rst calculate the pseudo-3D and/or pseudo-sur®cial parameters from the above mentioned. When all the surface pro®les have the same nominal length L, the pseudo-sur®cial parameter is obtained by calculating the arithmetical mean of the corresponding pro®le parameters. However, if the surface pro®les have di¨erent nominal lengths, it would be better to calculate a weighted mean. We consider that the fracture surface is made up of a set of ¯at and adjacent elementary facets that can be oriented in any direction in the space. Furthermore, we assume that any fracture surface de®ned by z ˆ f …x; y† is an element of C 2 (e.g. z, the ®rst and second derivatives of z are continuous). 2.1 Angularity Parameters 2.1.1 Pro®le Angularity A parameter, which di¨erently integrates the positive and negative slopes along a pro®le, was de®ned in order to take into account the shear direction. This procedure has already been carried out based on a statistical analysis of directional data (Riss and Gentier, 1990). This analysis consists of ®rst representing each angular datum of a pro®le by a unit vector making this angle, ai (or apparent colatitude), with the x- or y-axis in a trigonometric direction. Then, the mean direction ± that is the resultant of all the unit vectors (circular mean) ± is calculated. For this study, rather than to calculate the circular mean of the apparent colatitudes, we simply compute their arithmetical mean. However, for the directional data the arithmetical mean of the angular values is greater than their circular mean. Let Sp be the mean slope of the pro®le, Sp‡ the mean of positive slopes, and Spÿ the mean of negative slopes (Fig. 1). The mean angles of the inclinations of the pro®le …ÿ90 < yp ; yp‡ ; ypÿ < 90 † are calculated depending on the x or y axes by

Fig. 1. Slopes and angles of a topographic pro®le along x-axis

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the following relationships:   … L 1 dz…x† dx ; yp ˆ tan …Sp † ˆ tan L 0 dx    …  1 dz…x† ÿ1 ÿ1 ‡ ˆ tan …Sp † ˆ tan dx ; l…W‡ † W‡ dx ‡ ÿ1

yp ‡ and

ypÿ ˆ tanÿ1 …Spÿ † ˆ tanÿ1

ÿ1



1 l…Wÿ †



… Wÿ

dz…x† dx

 ÿ

 dx ;

…1† …2†

…3†

with L, the nominal length of the pro®le (Fig. 1); z, the height of the pro®le; W‡ and Wÿ , the set of all the intervals on which dz…x†=dx is respectively positive and negative; l…W‡ † and l…Wÿ † the total length of W‡ and Wÿ domains. For the regularly spaced data with a constant step Dx, Eq. (1) can be written in the discrete form as: ! N x ÿ1 X 1 z ÿ z i‡1 i …4† yp ˆ tanÿ1 …Sp † ˆ tanÿ1 ; Dx Nx ÿ 1 iˆ1 with zi , the algebraic values of heights along the pro®le; …Nx ÿ 1†, the number of intervals used for the slopes calculation; Dx, the sampling step along x-axis. The discrete form of the Eqs. (2) and (3) can be obtained by ®rst calculating the slope increments, Dz=Dx, along each pro®le and secondly the positive and negative mean of the slopes by these equations:  ! Mx ‡   X 1 Dz ; …5† yp‡ ˆ tanÿ1 …Sp‡ † ˆ tanÿ1 Mx‡ iˆ1 Dx ‡ i  ! Mx ÿ   1 X Dz ÿ1 ÿ1 ; …6† ypÿ ˆ tan …Spÿ † ˆ tan Mxÿ iˆ1 Dx ÿ i with Mx‡ and Mxÿ , the number of intervals on which are respectively calculated the positive slope …Dz=Dx†‡ and the negative slope …Dz=Dx†ÿ increments. During the course of a cyclic shear test along one direction (x or y), and for an interlocked joint, yp‡ will be taken into account in forward direction (for instance) and ypÿ in reverse direction. The weighted means of these parameters on the total number Np of the surface pro®les, correspond to the pseudo-sur®cial parameters …yp †k , …yp‡ †k and …ypÿ †k given by the following relationships (where k represents the x- or y-axis): 0 1 Np P B …Sp †kj Lkj C B C ÿ1 Bjˆ1 C; …7† …yp †k ˆ tan B N C @ Pp A Lkj jˆ1

Quantitative Parameters for Rock Joint Surface Roughness

0 B B …yp‡ †k ˆ tan B @

Np P

ÿ1 Bjˆ1

221

1 …Sp‡ †kj Lkj…‡† C C C; C N p P A Lkj…‡†

…8†

jˆ1

and

0 B B …ypÿ †k ˆ tan B @

Np P

ÿ1 Bjˆ1

1 …Spÿ †kj Lkj…ÿ† C C C; C N p P A Lkj…ÿ†

…9†

jˆ1

with …Lkj V 0†, the nominal length of the pro®le along the k-axis, Lkj…‡† ˆ Mk‡ Dk and Lkj…ÿ† ˆ Mkÿ Dk are the weight factors. 2.1.2 Surface Angularity To de®ne the 3D equivalent of yp , we assume that each surface is formed by an assembly of elementary ¯at surfaces de®ned by the topographical data …x; y; z† for each wall of the joint (Fig. 2a). Thus, it is possible to characterize the spatial orientation of each elementary surface from the azimuth and the inclination angle of its normal unit vector. The mean plane of each elementary surface (Fig. 2) is calculated by a linear regression of z ˆ f …x; y† with the least squares method. Then the inclination angle,

Fig. 2. Illustration of inclination angle of the elementary surfaces

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ak , of the normal vector of each elementary mean plane (angle between the normal to the regression mean plane and the vertical or z-axis). The three-dimensional or surface angularity is calculated from the ak angles of the normal vectors of all the elementary mean planes (Fig. 2b). The mean three-dimensional angle of the entire surface ys (0 U ys < 90 is calculated for the m normal vectors of the elementary mean planes according to: m 1X …ak †i : …10† ys ˆ m iˆ1 2.1.3 Surface Z2 Parameter In order to de®ne the three-dimensional equivalent of Z2, noted Z2s , we generalize the de®nition of Z2 by replacing the slope with the gradient norm of the surface heights. By assuming that the surface is continuum and di¨erentiating, let S be the actual area of the surface of the joint wall de®ned by the points M…x; y; z† so that x A ‰0; LxŠ, y A ‰0; L yŠ and z ˆ z…x; y† the heights with regard to the reference plane XOY . The Z2s parameter can be estimated by: " #1=2   # … Lx … Ly "  1 qz…x; y† 2 qz…x; y† 2 ‡ ; …11† Z2s ˆ dx dy Lx Ly 0 0 qx qy where Lx and Ly are the nominal lengths along the x and y axes; Lx Ly is the area of the reference plane XOY . Recall that for the linear pro®les, Z2 represents the square root of the quadratic mean of the gradient of z…x or y† or the mean slopes. In Eq. (11), Z2s represents also the root mean square of the slopes of the elementary surfaces that make up the entire surface. The discrete form of Eq. (11) can be approximated for all the elementary surfaces forming the surface by as follows: " " y ÿ1 x ÿ1 N X …zi‡1; j‡1 ÿ zi; j‡1 † 2 ‡ …zi‡1; j ÿ zi; j † 2 1 1 NX Z2s ˆ 2 2 …Nx ÿ 1†…Ny ÿ 1† Dx iˆ1 jˆ1 Ny ÿ1 N x ÿ1 X …zi‡1; j‡1 ÿ zi‡1; j † 2 ‡ …zi; j‡1 ÿ zi; j † 2 1 X ‡ 2 2 D y jˆ1 iˆ1

##1=2 ;

…12†

with Lx ˆ …Nx ÿ 1†Dx and Ly ˆ …Ny ÿ 1†D y; Nx the number of points along the x-axis; Ny the number of points along the y-axis; Dx and D y the sampling steps along the x and y axes; zi; j ˆ z…xi ; yj †. 2.2 Relative Roughness and Surface Tortuosity 2.2.1 Surface Roughness Coe½cient Reminder. Even if the linear roughness coe½cient, RL , is easily calculated, there remains a simple approximation of the desired surface roughness parameter which

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223

is the surface roughness coe½cient, noted Rs (Lange et al., 1993). Let At be the actual area of the surface of the fracture and An its nominal area. The nominal area is the projection of the fracture surface on its mean plane (which is parallel to the horizontal plane x y in this study) along the normal direction of the mean plane (Fig. 2a). El Soudani (1978) has de®ned the surface roughness coe½cient Rs as the ratio between the actual area At and the nominal area An : Rs ˆ

At An

with 1 U Rs :

…13†

As the shear plane can be more or less horizontal the Rs calculation requires that the surface heights zi; j be measured with respect to the mean plane. When Rs ˆ 1, the fracture surface is perfectly ¯at and smooth, and corresponds to the mean plane. The higher values of Rs are associated with the rougher surface. The coe½cient Rs measures in some way the degree of non-¯atness of the fracture surface with regard to the mean plane. Consequently, this coe½cient shows how much the fracture surfaces withdraw from or draw nearer to the mean plane. According to El Soudani (1978) the coe½cient Rs could be ranged from 1 to 5 when considering the set of fracture surfaces in metallurgy. The values of Rs < 2 correspond to the brittle fracture surfaces, and the values of Rs > 2 correspond to the ductile fracture surfaces with recovery. Kendall and Moran (1963) in El Soudani (1978) estimate the actual area At of a random surface as twice the nominal area An …At ˆ 2An †. El Soudani (1978) generalized this notion for any surface by the product of Rs and An : At ˆ Rs An . Calculation of the actual area of surfaces. Lange et al. (1993) suggest that the calculation of Rs requires that the actual area At of the fracture surface be directly measured (e.g. stereophotogrammetry images, scanning confocal microscope, atomic force microscope, etc.). With the fracture surface triangulation (Fig. 3), the authors estimate the actual area At of the fracture surface by summing the triangular element areas Ai as: X Ai : …14† At ˆ surface

Fig. 3. Triangulation of an elementary surface

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By carrying out topographic measurements of pro®les at small constant steps (e.g. 70 mm U Dx ˆ D y U 200 mm) with a laser sensor pro®lometer or other, we suggest that the actual area At of the fracture surface can also be evaluated by the integral method. Assuming that the fracture surface is continuum and derivable, its actual area At is given by the relationship: "  2  2 #1=2 … qz qz …x; y† ‡ …x; y† 1‡ dx dy: …15† At ˆ qx qy surface The discrete form of the Eq. (15) can be approximated for very small Dx and D y by: s     N y ÿ1 x ÿ1 N X X zi‡1; j ÿ zi; j 2 zi; j‡1 ÿ zi; j 2 ‡ 1‡ : …16† At A …DxD y† Dx Dy iˆ1 jˆ1 Joint interface roughness. The mechanical properties of rock joints do not depend on the roughness of a single wall but rather the sum of the heights forming the joint (Tsand and Witherspoon, 1983). The sum of the heights of the two walls designated as ``composite topography'' (which we call joint interface), is a function of the walls' degree of matching (Brown and Scholz, 1985). Consequently, each roughness parameter must be calculated for both each wall and the joint interface. As a reminder, the Rs parameter is formulated for a single wall surface. Knowing that the lower and upper joint walls are in contact, it seems better to quantify the joint interface roughness. Joint interface is de®ned by the composite topography or by the ``composite actual area'', Atc , which is the sum of the actual areas of the lower and upper wall surfaces …Atc ˆ Atl ‡ Atu †. The joint interface roughness is then calculated from the composite area Atc according to: Rs …joint† ˆ

Atc Atl ‡ Atu 1 l ˆ ˆ …Rs ‡ Rsu †; Anc 2An 2

…17†

where the exponent c represents the composite actual area, l and u correspond respectively to the areas of the lower and upper walls of the joint. When the two walls of the joint have the same dimensions, Anc ˆ 2An . For the brittle fractures the surface roughness coe½cient Rs will range from 1 to 2. 2.2.2 Surface Tortuosity Drawing on the sinuosity index, Ps , de®ned by Pikens and Gurland (1976), we de®ne its sur®cial equivalent which is the surface tortuosity coe½cient, Ts , as the ratio of the actual area At of the fracture surface and the area Ap of the surface formed by the four extreme points, delimiting the measurement zone (Fig. 4). For esthetic reasons, this surface will be designated as p-surface. For each fracture wall, the surface tortuosity coe½cient is given by the relationship: Ts ˆ

At At ˆ cos f ˆ Rs cos f Ap An

with 0 < Ts U Rs ;

0 < cos f U 1:

…18†

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Fig. 4. De®nition of the p-surface for the estimation of the area Ap

f is the angle between the normal of the p-surface and the normal of the mean plane of the joint surface. The p-surface will be ¯at only when the four extreme points are coplanar. Unfortunately, in most cases these four points will not necessarily be coplanar and consequently, this surface will probably not be ¯at. In practice, we recommend that the area Ap of the p-surface be estimated by calculating the mean plane, Pm , across the four extreme points (Fig. 4) by the least square regression method. The equation of this mean plane Pm will be in the form: Pm : ax ‡ by ÿ z ‡ g ˆ 0

and

1 cos f ˆ p : 2 a ‡ b2 ‡ 1

…19†

The surface tortuosity coe½cient Ts measures the degree of deviation of the fracture surface with respect to the mean plane Pm . When f ˆ 0 , Ts measures the degree of non-¯atness of the fracture surface with regard to the mean plane (as the Rs parameter). The tortuosity coe½cient of the joint interface is calculated using the composite area Atc as follows: Ts … joint† ˆ

Atc Atu ‡ Atl Atu ‡ Atl ˆ ˆ ; An An Apc Apu ‡ Apl u‡ cos f cos f l

…20†

where Apc , is the composite area of the p-surfaces, the exponents u and l designate respectively the upper and lower walls. For a perfectly interlocked joint, the inclination angle of the mean plane Pm of the upper wall, f u , is equal to the inclination angle of the lower wall, f l …f u ˆ f l ˆ f†, and the Eq. (20) becomes: Ts …joint† ˆ

Atu ‡ Atl Atu ‡ Atl ˆ cos f: 2Ap 2An

…21†

The more the Ts < Rs the more Ts ! 0 and the more tortuous the fracture surface. When Ts ˆ Rs , the surface is rough rather than tortuous. On the other

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hand, Ts ˆ 1 then cos f ˆ An =At ˆ 1=Rs . In this case either (i) Rs > 1 (rough surface), ÿ90 < f < 90 and the surface is tortuous or (ii) Rs ˆ 1 (smooth and ¯at surface), f ˆ 0 …cos f ˆ 1† and the surface is neither rough nor tortuous, but perfectly smooth and ¯at. Even though the surface tortuosity coe½cient Ts can allow to distinguish two di¨erent surfaces having the same Rs value.

3. Characterization of Primary Roughness The primary roughness is characterized in terms of real and apparent structural anisotropy of the surface. The real structural anisotropy is characterized by means of geostatistical analysis (variogram and auto-correlation function), while the apparent anisotropy is characterized with the aid of the linear parameters yp and Z2. For this study, we assume that the fracture surface satis®es the second order stationary hypothesis and, consequently, the surface heights must be measured with respect to its mean plane.

3.1 Real Structural Anisotropy of Surfaces In the literature, the experimental studies for the purpose of modeling the mechanical behavior of rock joints are most often carried out on various types of arti®cial mortar joints. The morphology of these joints being in saw teeth, irregular triangles, a combination of various triangles, undulations, etc.. But the common point between these surfaces is their anisotropy. Indeed, all these surfaces present di¨erent structures (or characteristics) along x and y directions (regular or irregular geometry). 3.1.1 Geostatistical Analysis In geostatistical sense a regionalized phenomenon is known as anisotropic if it presents particular directions of variability. But these privileged directions must correspond to genetic phenomena known a priori (Journel, 1975). The structural analysis of the 1D semi-variograms in all directions can reveal three types of behavior: 1. isotropic variability, where the semi-variograms have the same ``range value'' as well as the same ``sill'' 2. proportional e¨ect, where the semi-variograms have the same sill but di¨erent variances and 3. anisotropic variability, where the semi-variograms present the same global variability, in particular the same sill but with di¨erent range values. The variographic analysis showed that the range value a of the variogram and the correlation distance dc of the correlogram quantify the in¯uence zone limit of the studied phenomenon. Consequently, knowing their values in all directions allows to characterize the structural anisotropy of the studied surface. Rather than

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to calculate directional 1D semi-variograms or 1D correlograms, it is possible to obtain a cartographic representation (iso-value contours) of the incremental variances (2D variogramm or variomap) and the correlation (2D auto-correlograms or auto-correlomap) by relationships: g  …hx ; hy † ˆ

r…hx ; hy † ˆ

Mÿh Xx Nÿh Xy 1 …Zj‡hx ; k‡hy ÿ Zj; k † 2 2‰…M ÿ hx †…N ÿ hy †Š jˆ1 kˆ1

Mÿh Xx Nÿh Xy 1 …Zj; k ÿ Z†…Zj‡hx ; k‡hy ÿ Z† …M ÿ hx †…N ÿ hy † jˆ1 kˆ1 M X N 1 X …Zj; k ÿ Z† 2 MN jˆ1 kˆ1

;

…22†

…23†

with MN the total number of points on the surface; M ÿ hx and N ÿ hy the number of pairs of points respectively at a lag distance hx A ‰ÿM=2; M=2Š and hy A ‰ÿN=2; N=2Š; Zj; k the random variable representing the surface heights z…x; y†. Notice that with this kind of representation the shape of each iso-value (isovariance or iso-correlation) contour involves an anisotropy or an isotropy at a given scale of observation. An elliptic shape of the contour indicates a mean anisotropy, while a circular shape of this contour indicates a mean isotropy. 3.1.2 Method of Analysis of Real Structural Anisotropy In the author's point of view the only question that can be asked in the use of the 2D variogram or the 2D auto-correlogram in order to characterize the structural anisotropy is: which iso-value contour is representative of the whole surface structure? The answer is that for the 2D variogram (variomap) it is necessary to determine the range values in all directions and the corresponding iso-variance contour, while, for the 2D auto-correlogram, it is su½cient to consider the null isocorrelation contour. Indeed, one of the essential properties of the auto-correlation function, r…h†, is that the lag at which it becomes null …r…h†hˆdc ˆ 0† corresponds to the correlation distance dc. Beyond this distance the data are no longer correlated and are therefore random. This property makes possible and easy the direct description of the real structural anisotropy based on the null iso-correlation contour of the 2D autocorrelogram (Eq. 23). The 2D auto-correlogram or the correlation map is therefore the tool for the direct characterization of the real structural anisotropy. In this way, only the isocorrelation r…hx ; hy † ˆ 0 which exactly describes the structural anisotropy of the surface, will be considered (Fig. 5). When the null iso-correlation draws a circle or can be ®tted by a circle, the surface presents a structural isotropy. In contrast when this iso-correlation draws an ellipse or can be ®tted by an ellipse, the surface presents a structural anisotropy.

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Fig. 5. Typical 2D auto-correlogram showing the null iso-correlation

Fig. 6. Schematization of the anisotropy ellipse based on the 2D auto-correlogram

This method has already been used by Shaw and Smith (1990) for the description of sea¯oor topography with statistically heterogeneous data, but especially by Frykman and Rogon (1993) for the analysis of the anisotropy of porous networks. To characterize the real structural anisotropy of a surface using the null isocorrelation, we suggest to ®t it with an ellipse for which the half-large axis R and the half-small axis r are calculated so as to obtain the anisotropy ratio la ˆ R=r …1 U la †. By referring to ®gure 6 the half-large axis R, the half-small axis r and the principal direction angle of anisotropy, j, are calculated as:

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q 8 > R ˆ …XR † 2 ‡ …YR † 2 > > > > q   < 2 2 r ˆ …Xr † ‡ …Yr † : >   > > > Y > : j ˆ atan R XR

…24†

The principal direction of anisotropy corresponds to the direction of the lengthening of the ellipse and therefore to the mean direction of the surface large structures. For example, by taking an undulated surface that results in the fact that this principal direction is, on average, parallel to the orientation of the peaks of the undulations. For such surfaces, we therefore expect to have higher values of the linear parameters along the normal direction to the principal direction of anisotropy. Let us keep in mind the fact that the 2D auto-correlogram highlights the spatial organization of the structures forming the geometry of surface, while the linear parameters describes the geometry of this surface. 3.2 Apparent Anisotropy of the Surfaces Let Px and Py be one of the previous geometric linear parameters calculated respectively along the x-axis and the y-axis. These parameters can account for the apparent structural anisotropy of the surfaces. This apparent anisotropy can be described on the basis of the mathematical de®nition of an ellipse in the xoy coordinates system, with the half-axis a and the half-axis b (along x- or y-axis). To accomplish this, we de®ne the degree of apparent anisotropy, ka , as: b ka ˆ : a

…25†

According to the calculated linear parameters and the shape of the studied surface, Px and/or Py can take any values. So that the relationship (25) does not involved an indetermination, a must be di¨erent from zero and always greater than b. Consequently, we recommend that the half-axis a be always equal to maxfPx ; Py g and the half-axis b be always equal to minfPx ; Py g as ka ˆ

b minfPx ; Py g : ˆ a maxfPx ; Py g

…26†

After checking the parameter yp (weighted mean) has been retained to express the degree of anisotropy ka by the relationships: ka ˆ

minfypx ; ypy g maxfypx ; ypy g

with 0 U ka U 1:

…27†

So that the parameter ka describes the apparent structural anisotropy, we consider that the direction associated to the minfPx ; Py g value be qualitatively close to the principal direction of the real structural anisotropy (§3.1.2). Thus, when 0 U ka < 1, the surface is anisotropic (ka ˆ 0 corresponds to surfaces with

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saw teeth, undulated surfaces, etc.); when ka ˆ 1, the surface is isotropic. On the other hand, if ka ˆ Py =Px , the principal direction of anisotropy is parallel to the y-axis (transverse to the x-axis) and if ka ˆ Px =Py , the principal direction of anisotropy is parallel to the x-axis (transverse to the y-axis). The ka parameter exclusively quanti®es the anisotropy with regard to the x and y axes of the sample. Any deviation in relation to this coordinates system is not taken into account. As a matter of fact, for an anisotropic surface of which the direction of the lengthening of the ellipse of the real anisotropy makes an angle j ˆ 45 with the x-axis, the ka parameter will probably indicate an isotropy nature of the surface …Px ˆ Py †. As a result, this parameter is more suitable for the characterization of arti®cial anisotropic joints (saw teeth, undulations, . . .) and is very insu½cient for the analysis of structural anisotropy. 4. Application Examples 4.1 Experimental Procedures 4.1.1 Tested Samples For this study, four experimental series of direct shearing were conducted. The two ®rst series concern the man-made Lanhelin granite joints. These are noninterlocked and non-mated joints with ¯at sanded surfaces (sanding the walls under pressure) and hammered surfaces (slight hammering on the walls). The two last series concern the mortar replicas of a man-made regularly undulated surface and a natural schist joint with rough and undulated surface. These replicas were made from silicone moulds of the surfaces of about 145  150 mm 2 dimension by casting with mortar. The replicas of the regularly undulated surface are interlocked and mated, while those of the natural schist joint are only mated (due to the geological history of rock mass such as the e¨ect of weathering, the ¯uids ¯ow, the gouge material, the relative displacements, etc.). Three samples were sheared for series 1, three for series 2, six for series 3 and ®ve samples for series 4. All the man-made Lanhelin granite joints with sanded and hammered surfaces have the same dimension of 150 mm  150 mm, the same thickness of 40 mm and the same surface maximum amplitude of 1.742 mm. The mortar replicas have the same mean thickness of 40±45 mm, but a section of 100 mm  145 mm for the regularly undulated surface, and 135 mm  145 mm for the natural schist joint. The maximum amplitude is of 2 mm for the regularly undulated surface and of 8.103 mm for the natural schist joint. The properties of these three types of joint are listed in Table 1. 4.1.2 Direct Shear Experiments All the experiments were conducted at constant normal stress under di¨erent normal stress levels at a constant shear rate of 0.5 mm/mn. The three sanded joints and the three hammered joints underwent 5 cycles of direct shear under three normal stress levels ranging between 0.3±4 MPa; the six replicas of the man-made

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Table 1. Test sample properties No. of series

Material

Joint shape

Section (mm 2 )

sc (MPa)

st (MPa)

No. of sample

1 2 3 4

granite granite mortar mortar

ground surface sanded surface undulateda rough undulatedb

150  150 150  150 100  145 135  145

152 152 75 75

ÿ10 ÿ10 ÿ4 ÿ4

3 3 6 5

a Regularly undulated with period of 25 mm. b Natural schist joint replica.

regularly undulated joint undergone 10 cycles of shear under six normal stress levels ranging between 0.5±5 MPa and the ®ve replicas of the natural schist joint underwent a monotonous shear under ®ve normal stress levels ranging between 0.4±2.4 MPa. That is a total tangential displacement of 200 mm for the sanded and hammered joints, 400 mm for the regularly undulated joint, and 20 mm for the natural schist joint replicas. 4.1.3 Topography Data Acquisition In order to evaluate the contribution of the morphology to the mechanical behavior of sheared joints, topographical measurements were carried out before and after the shear tests with a laser sensor pro®lometer (Sabbadini et al., 1995; Homand-Etienne et al., 1995; Belem et al., 1997). This equipment allows threedimensional measurements of the joint wall surfaces (x, y, z coordinates). The measurement system uses the principle of laser triangulation between a laser plane and a CCD camera shifted with respect to the laser plane (the laser and video camera unit being non deformable). The topographic pro®le corresponds to the intersection of the laser beam with the sample surface. The laser pro®lometer is made up mainly of an optical sensor equipped with a CCD camera of a 50 mm resolution and with a He-Ne laser of 670 nm wavelength. The design features of the laser beam are: 40 mm length; 50 mm thickness; 50 mm of vertical resolution (z-axis); 73 mm of horizontal resolution (x- or y-axis according to the sensor position); 5 mm of standard deviation of the error of the white noise due to the mechanical vibration. Figure 7 shows a three-dimensional representation of the lower wall surfaces of the sanded and hammered granite joints, and Fig. 8 shows the lower walls of the mortar replicas of the natural schist joint and of the regularly undulated joint. 4.2 Calculation of the Morphological Parameters In the framework of this study, the man-made joints of sanded and hammered granite represent the ``purely'' secondary roughness (¯at joints), while the mortar replicas of the regularly undulated surface and the natural schist joint represent the combination of the secondary and primary components of the roughness. By

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Fig. 7. 3D surface plots of the Lanhelin granite: a sanded surface; b hammered surface

Fig. 8. 3D surface plots of the mortars: a schist joint replica; b regularly undulated surface

controlling all these aspects of the morphology, we hope to reach our objectives. To make a success of this study, a computer program, PARAM, has been developed to calculate morphological parameters. The calculated linear parameters are: the linear roughness coe½cient RL , the sinuosity index Ps , the RMS of the pro®le ®rst derivative Z2 and the pro®le angularities yp , yp‡ and ypÿ . The calculated sur®cial parameters are: the surface relative roughness coe½cient Rs and the surface tortuosity coe½cient Ts . The calculated three-dimensional parameters are: the three-dimensional angularity ys and the RMS of the surface ®rst derivative Z2s . The degree of apparent anisotropy ka is calculated from the parameter yp . In Tables 2 to 7, only the values of ys , Z2s and Rs are presented for the four types of tested joints. These values are calculated for the lower and upper walls, and for the joint interface before …sn ˆ 0† and after …sn 0 0† shearing. For the sanded and hammered joints, the shearing direction is along the y-axis and for the mortar replicas, the shearing direction is along the x-axis.

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Table 2. Calculated asperity angle ys for the granite joints sn (MPa)

0 0.3 1.2 3

Granite sanded surface joint ys (degree) lower wall

upper wall

joint interface

4.509 4.438 4.21 3.564

5.081 4.976 4.749 3.761

4.795 4.707 4.479 3.662

sn (MPa)

0 0.3 1.2 4

Granite ground surface joint ys (degree) lower wall

upper wall

joint interface

12.771 9.885 7.824 8.029

12.484 9.32 7.187 7.637

12.627 9.603 7.506 7.833

Table 3. Calculated asperity angle ys for the mortar replicas sn (MPa)

0 0.4 0.8 1.2 1.8 2.4 ±

Schist joint replica ys (degree)

sn (MPa)

lower wall

upper wall

joint interface

12.4 12.419 12.45 12.473 12.188 12.132 ±

11.352 11.181 11.508 11.274 11.32 11.386 ±

11.876 11.8 11.979 11.873 11.754 11.759 ±

0 0.5 1 2 3 4 5

Undulated joint replica ys (degree) lower wall

upper wall

joint interface

10.31 10.238 10.261 10.116 10.153 10.346 9.962

10.31 10.251 10.444 10.315 10.19 9.605 8.892

10.310 10.245 10.353 10.215 10.171 9.975 9.427

Table 4. Calculated Z2s for the granite joints sn (MPa)

0 0.3 1.2 3

Granite sanded surface joint Z2s lower wall

upper wall

joint interface

0.099 0.099 0.093 0.08

0.11 0.11 0.104 0.085

0.105 0.104 0.099 0.082

sn (MPa)

0 0.3 1.2 4

Granite ground surface joint Z2s lower wall

upper wall

joint interface

0.282 0.217 0.172 0.176

0.275 0.204 0.16 0.168

0.279 0.211 0.166 0.172

4.3 Joint Surfaces Anisotropy Equation (27) shows that when 0 U ka < 1, the surface is anisotropic and when ka ˆ 1 the surface is isotropic. With these ranges of variation, it is easier to point out anisotropy than isotropy because in a natural setting, it is less obvious to ®nd real isotropic surfaces. To determine an interval in which a surface can be considered as isotropic, we have calculated the apparent anisotropy coe½cients of the initial roughness of four types of studied joints. We then make the assumption that because of their manufacturing process, the sanded and hammered surfaces are

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T. Belem et al. Table 5. Calculated Z2s for the mortar replicas

sn (MPa)

0 0.4 0.8 1.2 1.8 2.4 ±

Schist joint replica Z2s

sn (MPa)

lower wall

upper wall

joint interface

0.334 0.335 0.325 0.319 0.322 0.313 ±

0.308 0.304 0.297 0.296 0.295 0.286 ±

0.321 0.319 0.311 0.307 0.308 0.3 ±

0 0.5 1 2 3 4 5

Undulated joint replica Z2s lower wall

upper wall

joint interface

0.214 0.208 0.209 0.201 0.204 0.219 0.215

0.214 0.209 0.212 0.204 0.203 0.196 0.18

0.214 0.208 0.210 0.203 0.204 0.208 0.197

Table 6. Calculated roughness coe½cient Rs for the granite joint sn (MPa)

0 0.3 1.2 3

Granite sanded surface joint Rs lower wall

upper wall

joint interface

1.0051 1.004 1.0046 1.0043

1.0065 1.0057 1.0042 1.0059

1.0058 1.0048 1.0044 1.0051

sn (MPa)

0 0.3 1.2 4

Granite ground surface joint Rs lower wall

upper wall

joint interface

1.0436 1.026 1.0161 1.0166

1.0416 1.0229 1.0137 1.0153

1.0426 1.0244 1.0149 1.0159

Table 7. Calculated roughness coe½cient Rs for the mortar replicas sn (MPa)

0 0.4 0.8 1.2 1.8 2.4 ±

Schist joint replica Rs

sn (MPa)

lower wall

upper wall

joint interface

1.0510 1.0508 1.0487 1.0475 1.0477 1.0457 ±

1.0432 1.0419 1.0416 1.0410 1.0407 1.0387 ±

1.0471 1.0464 1.0451 1.0443 1.0442 1.0422 ±

0 0.5 1 2 3 4 5

Undulated joint replica Rs lower wall

upper wall

joint interface

1.0225 1.0213 1.0217 1.0203 1.0211 1.0238 1.0227

1.0225 1.0214 1.0223 1.0208 1.0205 1.0194 1.0169

1.0225 1.0214 1.0220 1.0206 1.0208 1.0216 1.0198

considered as ``isotropic'' (i.e. the roughness parameters are quite the same order of magnitude along the x and y directions). Consequently, these roughness will be used as a basis for an arbitrary classi®cation of anisotropic morphologies. Based on the obtained results and in accordance with the initial assumption, we consider that when 0:9 U ka …yp † U 1, the surface can be regarded as isotropic. Beyond this interval, the surface is considered as more or less anisotropic. Table 8 presents

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Table 8. Anisotropy classi®cation Range of the variation of ka …yp †

Joint surface shape

0 U ka < 0:25 0:25 U ka < 0:5 0:5 U ka < 0:75 0:75 U ka < 0:9 0:9 U ka U 1

anisotropic surface more anisotropic than isotropic surface homogenous surface: neither anisotropic nor isotropic more isotropic than anisotropic surface isotropic surface

the di¨erent intervals of the rough classi®cation of anisotropic morphologies. It appears that the replicas of the schist joint and of the regularly undulated surface are strongly anisotropic. 4.4 Correlation Between Linear and Sur®cial Parameters Figures 9 and 10 compare the variations of the extended linear parameters (pseudosur®cial) and the three-dimensional and sur®cial parameters of the studied samples. Figure 9 presents the pairs of angularity parameters …yp…xy† ; ys † and …Z2…xy† ; Z2s †. Figure 10a presents the pair of roughness parameters …RL…xy† ; Rs † and Fig. 10b compares the tortuosity parameter to the sinuosity index …Ps…xy† ; Ts †. These results show that (i) the three-dimensional and sur®cial parameters are always greater than the extended linear parameters, (ii) the three-dimensional and sur®cial parameters vary in the same way as the extended linear parameters for a given sample surface. It is what explains the alignment of the data points in Figs. 9 and 10. By making assumption that the 3D parameters better characterize the joint surface roughness, the previous observation (i) involves the fact that the extended linear parameters underestimate the surface morphological features compared with the sur®cial and the 3D parameters. Observation (ii) con®rms or suggests the acceptability of the proposed parameters, which are not derived from the existing linear parameters. At ®rst approximation, the obtained results for the studied joints were ®tted to the linear models which are: ys A 1:6136yp…xy† ‡ 0:0325

with R ˆ 0:978 and

Stderror ˆ 0:551

…28†

Z2s A 1:5656Z2…xy† ÿ 0:0002

with R ˆ 0:985

and

Stderror ˆ 0:013

…29†

Rs A 2:3124RL…xy† ÿ 1:3138

with R ˆ 0:986

and

Stderror ˆ 0:002

…30†

Ts A 2:3215Ps…xy† ÿ 0:03229

with R ˆ 0:986

and

Stderror ˆ 0:002:

…31†

The average estimation di¨erences of the four linear parameters are about: …ys ÿ yp † A 3;681 , …Z2s ÿ Z2† A 0;080, …Rs ÿ RL † ˆ …Ts ÿ Ps † A 0;015. As all these simple relationships are obtained from the morphological data of two isotropic surfaces and two strongly anisotropic surfaces, these relationships can reasonably be used for any types of morphology. This correlation allows especially to obtain an approximation of the three-dimensional and sur®cial parameters from the linear parameters.

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a

b Fig. 9. Comparison between the pro®le and the surface angularities: a ys vs …yp †x ; …yp †y ˆ average of pro®le inclination angle yp…xy† ; b surface Z2s vs …Z2†x ; …Z2†y ˆ average of pro®les Z2…xy†

Figures 9 and 10 also show that: the more the slope of linear regression straight line is close to 1, the closer the linear parameters are to the three-dimensional and/ or sur®cial parameters. Figure 10a particularly shows that: the rougher the joint is (high Rs ), the more RL…xy† is underestimated. The variation of the slopes of linear regression straight lines directly points out the di¨erence in sensitivity of the parameters representing di¨erently the variations of the same morphology, except the parameters Rs and Ts , which are quite close when the surface is not tortuous. However, the value of the slope of regression straight involves the magnitude of the underestimation of the linear parameter. 4.5 Evolution of the Morphology During the Course of Shearing The evolution of the sheared joints roughness has been quanti®ed by the 3D angularity parameter ys and the surface relative roughness coe½cient Rs (Figs. 11

Quantitative Parameters for Rock Joint Surface Roughness

237

a

b Fig. 10. Comparison between: a extended linear …RL †x ; …RL †y and surface roughness coe½cients Rs ; b extended sinuosity index …Ps †x ; …Ps †y and surface tortuosity coe½cient Ts

and 12). The results overall show that (i) after 5 cycles of shearing, the roughness of the sanded granite joints decreases gradually with the normal stress level, (ii) a strong reduction in the roughness of the hammered granite joints with the increase of the normal stress level; this reduction tends to be stabilized at high stress levels (sn ˆ 4 MPa), (iii) the roughness of the mortar replicas of the natural schist joint decreases slightly and continuously with shearing and under various normal stress levels, (iv) after 10 cycles of shearing the roughness of mortar replicas of the regularly undulated surface decreases slightly with normal stress.

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Fig. 11. Variation in ys with the normal stress sn

Fig. 12. Variation in Rs with the normal stress sn

5. Conclusion Several morphological parameters have been de®ned in order to better characterize the secondary roughness (roughness sensu stricto) and the primary roughness (anisotropy), which seems to play a large role in the dilatant behavior of rock joints during the course of shearing. The primary roughness was quanti®ed by the degree of apparent anisotropy ka which is de®ned by the means of the linear parameter yp calculated along two directions (x and y). The secondary roughness was quanti®ed by the parameters calculated from all the surface pro®les according to bi- and three-dimensional approaches. Thus, the three-dimensional angularity

Quantitative Parameters for Rock Joint Surface Roughness

239

is quanti®ed by the mean surface angle, ys , and the RMS of the ®rst derivative, Z2s . This roughness is also quanti®ed by the surface roughness coe½cient Rs (El Soudani, 1978) on the basis of the estimation of the actual areas of the surfaces (developed areas) by the triangulation and the integral methods. These parameters were then used to characterize the evolution of the morphology during the course of the shearing of man-made sanded and hammered granite joints (5 cycles of shearing under three normal stress levels), mortar replicas of a natural schist joint (monotonous shearing) and mortar replicas of a regularly undulated surface (10 cycles of shearing). From these results it comes out that (i) the linear parameters are always lower than the 2D and 3D parameters and as a result underestimate the characteristics of the morphology, (ii) the proposed 2D and/or 3D parameters quantify the same things as the traditional linear parameters, and (iii) the angularity parameters best represent the geometry of surfaces and are more suitable for the analysis of the anisotropy of surfaces. In work which will be published soon we will use all these parameters to directly characterize the e¨ective degradation of rock joints during the course of shearing. This will be followed by a modeling of the evolution of peak dilatancy and state of a joint subjected to any path of shearing. These models will be used for the de®nition of a failure criteria which will be taken into account in a constitutive law of rock joints.

List of symbol and parameters Parameters associated with linear pro®les in x-direction: x: distance from origin z; z…x†: height of the pro®le from the reference base line (e.g. amplitude of asperity height at the distance x) zi ˆ z…xi † i ˆ 1; N: discrete values of asperity height L or Lx : nominal pro®le length Dx: small constant distance between two adjacent amplitude readings (constant step in x-direction) dz…x† : elementary pro®le slope (e.g. ®rst derivative of z…x† with respect to x) estidx Dz z…x ‡ Dx† ÿ z…x† ˆ mated by Dx Dx   Dz tanÿ1 : elementary pro®le inclination angle Dx s   … xˆL 1 dz…x† 2 1‡ dx: linear roughness coe½cient RL ˆ L xˆ0 dx RMS: Root Mean-Square s    … 1 xˆL dz…x† 2 dx: RMS of ®rst derivative of the pro®le Z2 ˆ L xˆ0 dx

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   zN ÿ z1 Ps ˆ RL cos tanÿ1 : sinuosity index L Sp : mean of the absolute values of pro®le slopes yp ˆ tanÿ1 Sp : mean angle of the pro®le inclination angles Sp‡ : mean of the positive values of pro®le slopes yp‡ ˆ tanÿ1 Sp‡ : mean of positive pro®le inclination angles Spÿ : mean of the negative values of pro®le slopes ypÿ ˆ tanÿ1 Spÿ : mean of negative pro®le inclination angles Dz are positive W‡ : set of intervals where the elementary pro®le slopes Dx l…W‡ †: length of W‡ Dz are negative Wÿ : set of intervals where the elementary pro®le slopes Dx l…Wÿ †: length of Wÿ    Nx : total number of pro®le points Dz Mx‡ : number of intervals in domain W‡ e:g: number of Dx   ‡  Dz Mxÿ : number of intervals in domain Wÿ e:g: number of Dx ÿ Lxj…‡† ˆ Mx‡ Dx: length of j th interval of W‡ Lxj…ÿ† ˆ Mxÿ Dx: length of j th interval of Wÿ Lxj ˆ Lxj…‡† ‡ Lxj…ÿ† : nominal length of j th pro®le in x-direction Pseudo-sur®cial parameters: Let k be the x or y-axis …Sp †k : weighted mean of Sp for all pro®les in k-direction …yp †k : corresponding angle of …Sp †k (e.g. ``average'' of inclination angles with respect to k-direction …Sp‡ †k : weighted mean of positive slopes for all pro®les in k-direction …yp‡ †k : corresponding angle of …Sp‡ †k …Spÿ †k : weighted mean of negative slopes for all pro®les in k-direction …ypÿ †k : corresponding angle of …Spÿ †k …RL †k : weighted mean of RL for all pro®les in k-direction …Z2†k : weighted mean of Z2 for all pro®les in k-direction Surface parameters: M…x; y; z†: points de®ning the joint surface z ˆ z…x; y† ˆ f …x; y†: height of the joint surface from the reference base plane (e.g. amplitude of asperity height at the distances x and y) qz…x; y† qz…x; y† ; : partial derivative of z…x; y† with respect to x and y, respectively, qx qy for a given y or x zij ˆ z…xi ; yj † i ˆ 1, Nx ; j ˆ 1; Ny : discrete values of asperity height Lx Ly : area of the reference base plane (e.g. mean plane through joint surface points called surface mean plane)

Quantitative Parameters for Rock Joint Surface Roughness

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ka : apparent anisotropy degree g  …hx ; hy †: 2D variogramm r…hx ; hy †: 2D auto-correlogram hx ; hy : distance between points in x and y direction, respectively r…h†: auto-correlation function dc : correlation distance R: half-large axis of the ellipse corresponding to null iso-correlation contour r…hx ; hy † ˆ 0 r: half-small axis of the ellipse corresponding to null iso-correlation contour r…hx ; hy † ˆ 0 j: direction of anisotropy At : actual area of the joint wall surface An : nominal area (e.g. projection of At on its mean plane) Atu : actual area of the upper joint wall Atl : actual area of the lower joint wall Atc ˆ Atu ‡ Atl : composite actual area

Anc ˆ Anu ‡ Anl : nominal composite area Rs : surface roughness coe½cient Ts : surface tortuosity coe½cient p-surface: surface formed by the four extreme points of the studied joint surface Ap : area of p-surface Pm : p-surface mean plane f: angle between the normal of p-surface and the surface mean plane ones f u and f l : angle f of the upper and lower joint wall p-surface, respectively ak : elementary inclination angle (e.g. angle between the normal of an elementary surface and the normal of reference base plane) ys : mean of angles ak over the whole surface v u    ! u 1 … xˆLx … yˆLy qz…x; y† 2 qz…x; y† 2 t ‡ dx dy: RMS of the joint Z2s ˆ Lx Ly xˆ0 yˆ0 qx qy surface gradient References È ., Shimizu, Y., Kawamoto T. (1996): The anisotropy of surface morphology Aydan, O characteristics of rock discontinuities. Rock Mech. Rock Engng. 29(1), 47±59. Belem, T. (1997): Morphologie et comportement meÂcanique des discontinuiteÂs rocheuses. TheÁse de Doctorat INPL, Nancy, 220pp. Belem, T., Homand-Etienne, F., Souley, M. (1997): Fractal analysis of shear joint roughness. Int. J. Rock Mech. Min. Sci. 34(3±4), No. 130, 10p. Brown, S. R., Scholz, C. H. (1985): The closure of random elastic surfaces in contact. J. Geophys. Res. 90, 5531±5545. El Soudani, S. M. (1978): Pro®lometric analysis of fractures. Metallography 11, 247±336. Frykman, P., Rogon, T. A. (1993): Anisotropy in pore networks analyzed with 2-D autocorrelation (variomaps). Computers Geosci. 19(7), 887±930.

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Homand-Etienne, F., Belem, T., Sabbadini, S., Shtuka, A., Royer, J.-J. (1995): Analysis of the evolution of rock joints morphology with 2D autocorrelation (variomaps). In: Favre and Mebarki (eds.), Proc. 7th Int. Conference on Appl. Stat. & Proba, Paris, Lemaire. Balkema, Rotterdam, 1229±1236. Jing, L., Nordlund, E., Stephansson, O. (1992): An experimental study on the anisotropy and stress-dependency of the strength and deformability of rock joints. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 29, 535±542. Journel, A. (1975): GeÂostatistique minieÁre. Centre de GeÂostatistique de Fontainbleau, 308p. Kana, D. D., Fox, D. J., Hisiung, S. M. (1996): Interlock/friction model for dynamic shear response in natural jointed rock. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 33(4), 371±386. Lange, D. A., Jennings, H. M., Shah, S. P. (1993): Relationship between fracture surface roughness and fracture behaviour of cement paste and mortar. J. Am. Ceram. Soc. 3, 589±597. Pikens, J. R., Gurland J. (1976): Metallographic characterization of fracture surface pro®les on sectioning planes. 4th Int. Congr. Stereol., Gaithersburg. Riss, J., Gentier, S. (1990): Angularity of a natural fracture. In: Proc. MJFR, Rossmanith, H.-P. (ed.), Balkema, Rotterdam, 399±406. Riss, J., Gentier, S., Archambault, G., Flamand, R., Sirieix, C. (1995): Irregular joint shear behavior on the basis of 3D modelling of their morphology: Morphology description and 3D modelling. In: Rossmanith, H.-P. (ed.), Proc. MJFR, Balkema, Rotterdam, 157±162. Sabbadini, S., Homand-Etienne, F., Belem, T. (1995): Fractal and geostatistical analysis of rock joints roughness before and after shear tests. In: Rossmanith, H.-P. (ed.), Proc. 2nd Int Conf. on Mech. of Jointed and Faulted Rocks, Vienna. Balkema, Rotterdam, 535± 541. Shaw, P. R., Smith, D. K. (1990): Robust description of statistically heterogeneous sea¯oor topography through its slope distribution. J. Geophys. Res. 95(B6), 8705±8722. Tsang, Y. W., Witherspoon, P. A. (1983): The dependence of fracture mechanical and ¯uid ¯ow properties on fracture roughness and sample size. J. Geophys. Res. 88(B3), 2359± 2366. Wan, R. G., Achari, G., Schacter, R., Joshi, R., Mclellan P. J. (1996): E¨ect of drilling ¯uids on shear strength properties of Fernie shales. In: Aubertin, M., Hassani, F., Mitri, H. (eds.), Rock mechanics: Tools and techniques. Balkema, Rotterdam, 973±979. Yoshioka, N., Scholz, C. H. (1989): Elastic properties of contacting surfaces under normal and shear loads, 1, Theory. J. Geophys. Res. 94, 17681±17690. Authors' address: Prof. FrancËoise Homand, Laboratoire Environnement GeÂomeÂcanique et Ouvrages, Ecole Nationale SupeÂrieure de GeÂologie, rue du Doyen Marcel Roubault, BP 40, F-54501 Vandoeuvre-les-Nancy Cedex, France.