International Journal of Rock Mechanics & Mining Sciences 40 (2003) 947–954

Technical Note

An experimental method to link morphological properties of rockfracture surfaces to their mechanical properties

P. Lopeza,b,c,*, J. Rissa, G. Archambaultb

aCentre de D!eveloppement des G!eosciences Appliqu!ees (CDGA), Universit!e Bordeaux 1, Avenue des facult!es 33405 Talence Cedex, FrancebCentre d’!etudes en ressources min!erales (CERM), Universit!e du Qu!ebec "a Chicoutimi, 555 Bld. de l’Universit!e, Chicoutimi, P.Q., CanadacLaboratoire Environnement G!eom!ecanique et Ouvrages (LAEGO), !Ecole des Mines de Nancy, Parc de Saurupt, 54002 Nancy, France

Accepted 1 April 2003

1. Introduction

Experimental shear tests on rock fracture surfacesshow that understanding the shear behaviour needs totake into account the surface morphological aspects [1].During the past 30 years, there have been several

attempts to introduce the morphological aspects in rockjoint shear modelling, but these studies are restricted tosome part of the stress–strain–dilatancy behaviour. Infact, the shear behaviour can be divided into five phases[2,3] showed in the shear stress–shear displacementcurves (Fig. 1). Classically, the third phase (III in Fig. 1),i.e. the peak shear strength, was the most studied phase.It was also the first to be analysed [4–6]. The studiescarried out by Patton [4] and Ladanyi and Archambault[5] used sawtooth irregularities morphology whileBarton [6] used profiles on irregular joint surfaces. Qiu[7] developed an incremental model for the whole shearstress–shear displacement phases (I–V) using a sinetooth morphology for irregularities. A recent study,Flamand [8], used a different model for each phase ofthe shear behaviour.Actually, there is no existing model that integrates the

real morphology of rock fracture surfaces, usingmorphological factors (term used for parameters inexperimental design method) to represent it over thewhole shear displacement.This paper briefly reviews existing mechanical models.

Then, the results obtained from an experimental design

method using morphological and few mechanical factorsare presented in details.

2. Existing mechanical models

The LADAR model, Ladanyi and Archambault [5], isa theoretical approach for the peak shear strength usinga sawtooth morphology. It postulates, according toRowe et al. [9], that the peak shear strength is the sum ofthree friction—a component due to the shear failure ofasperities acting on the As (damaged area) projectedsurface and the dilatancy work components acting onthe A � As projected surface. They are expressed as afunction of the mean applied stresses on the total jointsurface A by the following equation:

tp ¼sð1� asÞð’v þ tan fmÞ þ astca

1� ð1� asÞ’v tan fm; ð1Þ

where tp is the peak shear strength, A is the total jointsurface, As is the sheared surface, as ¼ As=A is thesheared area ratio, s is the mean applied normal stress, ’vis the dilatancy rate, fm is the basic friction angle on thejoint wall surfaces and tca is asperities shear strengthgiven by any failure criteria.For as and ’v parameters, Ladanyi and Archambault

[5] proposed empirical equations.Gerrard [10] has tested the capabilities of six models

for respecting physical conditions and being able torepresent the shear behaviour of rock fracture surfaces.The LADAR model satisfies all the conditions exceptthe expression of as versus normal stress (s), that doesnot represent adequately the physical aspects. He alsoshows that it is the only model that expresses thephysical phenomena in analytic form. More recently,Flamand [8] linked the LADAR asperity angle io with

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*Corresponding author. Laboratoire Environnement G!eom-!ecanique et Ouvrages (LAEGO), !Ecole des Mines de Nancy, Parc de

Saurupt, 54002 Nancy, France.

E-mail addresses: philippe.lopez@mines.inpl-nancy.fr (P. Lopez),

j.riss@cdga.u-bordeaux.fr (J. Riss), guy archambault@uqac.ca

(G. Archambault).

1365-1609/03/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S1365-1609(03)00052-2

the statistical parameters of 2D angularity distributiony2 of asperities and characterised as and ’v in relationwith these statistical parameters.The JRC-JCS empirical joint peak shear strength

model of Barton [6] used profiles of irregular surfacemorphology and is summarised in the followingequation Goupy [23]:

t ¼ sn tan JRC log10JCS

sn

� �þ fb

� �; ð2Þ

where t is the peak shear strength, sn is the effectivenormal stress, JRC is the joint roughness coefficient,JCS is the joint wall compressive strength (determinedby a Schmidt Hammer test) and fb is the basic frictionangle (obtained from residual shear tests on flatunweathered rock surfaces).The JRC can be determined by a visual or a

computing method. The first one is quite subjective,and sometimes can give very different fracture strengthvalues (Tse and Cruden [11]). This estimation is, thus,inaccurate and questionable (Yu and Vayssade [12]).The second one results from tests (tilt tests strengthcompression tests) and computations. A large number ofstudies focused on trying to express the JRC versusmorphological factors such as Z2; RMS, by simpleregression equations, but these equations are only validfor the experimental case that were determined by Yuand Vayssade [12] and Lamas [13].What does the JRC represent? Barton presents this

parameter as a morphology parameter. It has an angledimension according to Barton and Choubey [14] but itcannot be assumed that it is a morphological factor: asoft rock and a hard rock do not have the same JRCvalue even if they have the same morphology. Therefore,the JRC is both a mechanical and a morphologicalfactor (Bougnoux [15]).Later on, Barton and Bandis [16] have integrated the

scale effect and asperities shear failure in their modelusing an angle parameter that is a questionable form(Gerrard [10]). Even if this model is easy to use, it still

does not take into account the physical aspects: itdescribes experimental curves and does not represent themechanical processes and the morphology of thefracture surfaces (Bougnoux [15]).The a2g model of Qiu [7] is an incremental

constitutive model using a sine tooth morphology. Thismodel is based on numerous assumptions not presentedhere. Several of them were taken from the theory ofplasticity (Lemaitre and Chaboche [17]), and lead to theexpression of plastic deformation, that depends on thefracture surface microstructure (a; contact point irregu-larities angle), and on the surface damage mechanism.This model assumes a decreasing thickness of the joint

caused by asperities destruction, and increasing of thisthickness caused by broken asperities material accumu-lation. The result is a formulation of the elasto-plastictensor depending on sliding and dilatancy.Modelled numerical results were compared by Qiu [7],

Huang et al. [18] tests performed on laboratory speci-mens that respect the a� g model assumptions. Qiu [7]found good agreements between both, but concluded,that the principal modification to be done, is to integratethe morphology closer to natural fracture surface.All these models and other studies performed during

the past three decades were trying to integrate repre-sentative joint surface morphology parameters withintheir modelling, but only partially, and anyway not for anatural fracture surface.An experimental design method is used to solve, at

least partly this problem (e.g. introduce the morphologycharacterisation in a mechanical behaviour law) andintegrate the five phases into one model. The morphol-ogy is simulated with the aim of evaluating themorphological factors.

3. Evaluation of morphological factors

A series of shear tests were conducted on identicalreplicas of the two walls of a natural fracture in a granite(Gu!eret, Creuse, France), with a constant rate of sheardisplacement (Fig. 2). The normal stress (s) and the sheardisplacement (U) are imposed and are called themechanical factors. The shear stress (t) and the normal

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Fig. 2. Direct shear tests.

Fig. 1. Five phases of the shear behaviour under three different

normal stresses (after [2,3]).

P. Lopez et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 947–954948

displacement (V ) are measured and called the responses.The data available are from morphological and mechan-ical aspects:

* The available data are a set of points measured alongprofiles on the original joint walls surfaces byFlamand [8], as shown in Fig. 3.

* The responses are continuous values of t and V

recorded during the tests.

A geostatistical simulation (kriging), Matheron [19], isapplied to estimate the morphological factors fromprofile measurements (x; y; z). The simulation is based

on the analysis of the variograms showing the morpho-logical anisotropy of the fracture walls (Fig. 4). Thevariograms show structures in the four directions, withthe same range but different sills. Our model took thatanisotropical information into account. The result ofsuch a simulation is a set of points (x; y; z) at the nodesof a regular grid that can be represented as in Fig. 5.Then the followings morphological factors were com-puted in every direction (between 0� and 359�, degree bydegree): moments and central moments of heights (Z) oforder 1–4, the range, CLA; RMS; RL; Z4; RA; Z2; Z3;coefficient of concentration (CC), skewness and kurtosiscoefficients, and the angularity (y2) factors. The defini-tions of these factors can be found in Lamas [13],Gentier [20] and Dagn!elie [21], and informations thatresults from computing these factors are developed inLopez [22].

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60°

0°

90°

-30°

-60

-40

-20

0

20

40

60

0

X (mm)

Y (mm)

120

Fig. 3. The whole set of profiles on a specimen.

90°

0°

-30°

60°

0.0

0.0

0.00 0.00

0.25 0.25

0.50 0.50

0.75 0.75

1.00 1.00

1.25 1.25

γ (h

)m

m2

Z

5.0 10.0 15.0 20.0 25.0 30.0

5.0 10.0 15.0 20.0 25.0 30.0

mm

Fig. 4. Four direction variograms: morphological anisotropy of the

fracture walls.

Fig. 5. Simulation (kriging) of a specimen.

P. Lopez et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 947–954 949

Some of these coefficients are global and notconcerned with profile directions such as moments andcentral moments of heights (Z) of order 1–4, the range,CLA, RMS, while others are related to profile direction(RL; Z4; Z3; Z2; y2).The matrix of observations including 360 lines, the

directions, and five columns, the five previous factors, isobtained. These data are analysed with classical statisticsand two multidimensional statistics: Principal ComponentAnalysis (PCA) and cluster observations (Lopez [22]). As aresult of this statistical analysis, four independent factors(RL; Z4; Z3; y2) (Table 1) were selected as inputmorphological data in an experimental design method.

4. Application of experimental design method

This method establishes relationships between thevarious factors characterising a problem with a mini-mum number of tests. Usually the elaboration of anexperimental design is performed before any test isperformed. However, in a few cases, like this study, theapplication of this method on existing tests is possible.The aim here is to find out what are the main factors,morphological and mechanical, that explain the shearbehaviour of a rock joint. Finding causation betweenthe different factors is another study.The direct shear tests performed under CNS (Con-

stant Normal Stress) by Flamand [8] are used in thisstudy (Fig. 2).A classical experimental design elaboration followed

Goupy [23], Schimmerling et al. [24], and Benoist et al.[25]. Details about the construction of the experimentaldesign can be found in Lopez [22].

Classically, an experimental design is presented as ablack box (Fig. 6), with input data (the factors) andoutput data (the responses). The question is how couldwe model what is happening in this black box in order toget an analytical expression that explains the mechanicalbehaviour of the joint (black box), with mechanical andmorphological factor values.Among the three levels of CNS used by Flamand [8],

two of them (7 and 21MPa) are introduced as inputdata and one is kept for validation. As four stress levelsare needed for the model computation, values ofmechanical responses are inferred for two others stresslevels (12 and 16MPa), see Lopez [22].The design of experiment is summarised hereafter:

* factors: s; U ; {RL; Z4; Z3 or y2},* responses: t and V ;* factor levels (factor values): see Table 2,* chosen model with 16 parameters: see Eq. (3). Thewriting of the model for the V response is similar.

The matrix form of Eq. (3) is given in Eq. (4). Tosimplify the writing of the matrix form, the terms with

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Table 1

Morphological factor definition

RL ¼Lt

L¼

True length

Project length

Linear roughness

Z4 ¼

PðDxiÞp �

PðDxiÞn

L

Proportion of the positive length of profile diminishes by the negative

length of profile divided by the total length of profile

Z3 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

L

R L

0

d2z

d2x

� �2

dx

sE

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

ðN � 2ÞðDxÞ4PN�1

i¼2 ðziþ1 � 2zi þ zi�1Þ2

s Root mean square of the square of the second derivative of the height

y2 2D colatitudes

Factors Responses

Law ?

Fig. 6. Illustration of an experimental design.

Table 2

Factor levels

Factor Low level High level Number of levels and levels

s 7 21 4 levels (7, 12, 16 and 21MPa)

U 0 5 11 levels (from 0 to 5 step 0.5mm)

RL 1.013 (0�) 1.024 (90�) 4 levels (1.016, 1.013, 1.020, 1.024)

Z4 �0.115 (�30�) 0.244 (90�) 4 levels (�0.115, �0.007, 0.189, 0.244)Z3 0.261 (0�) 0.335 (90�) 4 levels (0.330, 0.261, 0.324, 0.335)

y2 6.732 (0�) 10.267 (90�) 4 levels (6.834, 6.732, 9.496, 10.267)

P. Lopez et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 947–954950

an exponent greater than two do not figure. The 1 and�1 in Eq. (4) are the reduced and centred values of thefactors: �1 is the minimum value and 1 is the maximumvalue. Other values are intermediate.

t ¼ a0 þ a1RL þ a2sþ a3U þ a12RLsþ a13RLU

þ a23sU þ a11R2L þ a22s2 þ a33U

2 þ a333U3

þ a3333U4 þ a33333U

5 þ a333333U6

þ a3333333U7 þ a33333333U

8; ð3Þ

This equation is solved with the minimum residualmethod. The validation of the model is based on theanalysis of some statistical parameters, computed afterthe resolution of this equation, that can be found inGoupy [23]: sum of squares of predicted responses,residual sum of squares, residual standard deviation, R2;R2adjusted; degrees of freedom number (ddl), Fisher–

Snedecor test factor (F ) and residual diagrams.

5. Results and discussion

These parameters can be used to quantify theinfluence of the different factors on the responses, forthe case of our study they give a quantification of thequality of the model, using one of the four morpholo-gical model. The following four model configurationswere studied: (RL; s; U), (Z4; s; U), (Z3; s; U) and (y2;s; U). Among them, the one with Z4 stands out: theresults obtained from (Z4; s; U) show a good agreement

between tests and model. The parameters R2 andR2adjusted reach, respectively, 0.98 and 1.00 for dilatancy

V ; and 0.87 and 0.94 for shear stress t: Fig. 6 confirmsthe good agreement between test and model resultsrepresenting dilatancy V : The model accuracy permitsto simulate the contractancy phase of this response.Fig. 7 shows that the whole five classical phases for

the mechanical behaviour of a rock joint are wellsimulated using just one model. Among all advantagesof this model, the capability of representing the five

phases of the shear stress–shear displacement of thejoint mechanical behaviour (Archambault et al. [2] andArchambault et al. [3]), is the most important.The V response is obtained with a better agreement

than the t response regarding the test values (Fig. 8).This is especially the case at the origin and at the peak ofthe t curve. At the origin, the physical results must givezero for both responses (t and V ). The multilinearregression used, implicit in an experimental designmethod, do not impose these points: (U ¼ 0; t ¼ 0)and (U ¼ 0; V ¼ 0) whatever are the values of s and themorphological factors. To take into account these valuesthe structure of the multilinear regression has to bechanged. This work is planned to be done.For this model (Z4; s; U) the only difference between

the shear stress t and dilatancy V responses is that theinteraction Z4 *U does not affect the response of t: Inother words, an interaction in which a factor of lengthdimension (U in Z4 *U) influences only the response witha length dimension (V ), and not the stress response (t).

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(4)

P. Lopez et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 947–954 951

The model (Z3; s; U) also gives good information.The R2 and R2

adjusted values are 0.63 and 0.75 for V and0.67 and 0.90 for t; respectively. Furthermore, the Z3

factor influences neither the responses of t nor V : The

interactions Z3 *s and Z3 *U do not affect theresponses, and this model shows clearly that Z3 is notrequired within a joint shear behaviour simulation.This result must not to be taken the wrong way:

according to a recent work (Marache et al. [26]), Z3 isstill a factor conditioning the geostatistical simulationsbecause it expresses a morphological aspect of the rockfracture surfaces in general.Hence, we showed that the Z3 factor has no

information that influence the mechanical behaviour ofthe rock joint, but still conditioning a geostatisticalsimulation.The (RL; s; U) and (y2; s; U) model results show a

good agreement between tests and model values. Eachone of these two factors explains a part of themorphology: this is just a confirmation that RL isexplaining the ‘‘height part’’ of the morphology and y2 isexplaining the ‘‘angularity part’’ of the morphology.Therefore, it is largely demonstrated that, among the

four chosen factors, Z4 is the more accurate to intervenein the simulation of the mechanical behaviour of thisrock joint. Figs. 9 and 10 show how the mechanical andmorphological aspects are linked in this parameter.Fig. 11 illustrates the computations of the shear andnormal stresses on a projected surface of the wholesurface. Besides that, the Z4 factor has directionalinformation: for example, its sign changes if calculatedin the opposite direction (Fig. 10). For each facet ‘‘s’’(Fig. 10), only the area in contact will be considered: theZ4 expression is given by Eq. (5). It shows how onlythe ‘‘s’’ areas are contained in the positive part of thecontact surface.

Z4 ¼

PðDxiÞp �

PðDxiÞn

L; ð5Þ

where Dxi are the lag measures in the orientation ofdisplacement, p is the subscript letter for facet in theorientation of displacement (‘‘+’’ in Fig. 10), n is thesubscript letter for facet in the opposite orientation ofdisplacement (‘‘�’’ in Fig. 10).Another result concerns the quality of the prediction

that is given by the model. A test for a possibleimprovement of the model of t is performed. To do thiswe could give more information to the experimental

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-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0

V (

mm

)Exp 7MPa7 MPa 60°Exp 11,67 MPa11,67 MPa 60°

Exp 16,33 MPa16,33 MPa 60°Exp 21 MPa

21 MPa 60°

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

U (mm)

Fig. 7. Dilatancy, model/tests at 60�, sequence 2 (with morphological

factor Z4).

-5

0

5

10

15

20

25

30

0

U (mm)

τ (M

Pa)

Exp 7MPa 7 MPa 60°

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Exp 11,67 MPa

Exp 16,33 MPa

Exp 21 MPa

11,67 MPa 60°16,33 MPa 60°21 MPa 60°

Fig. 8. Shear stress, model/tests at 60�, sequence 2 (with morpholo-

gical factor Z4).

F

S

σ

Fig. 9. Actual computation of the stresses.

P. Lopez et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 947–954952

design and therefore to the model in considering anotherdistribution of the U levels: 0; 0.25; 0.5; 0.75; 1; 1.25:1.5; 2; 3; 4; 5. The aim here is a better simulation of thepeak phase of the t curves.The result is a less predictive model for the t response,

regarding the factors and interactions that influence itsresponse. Well-distributed levels are a recurrent condi-tion in experimental design method. In fact, there is afunction called the prediction function (6) that gives thevalues of the prediction errors using a model Goupy [23].

dð #yiÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixt

iðX tX Þxi

q: ð6Þ

This is another point favourable to the Z4 factorcompared to the three other morphology factors; itslevels are the closest to a uniform distribution.The last result from this experimental design method

is the Mohr peak shear strength envelopes, and they canbe deduced from this model. For example, the peakshear strength envelope is computed in finding themaximum values of t for each s values, Fig. 11. So,considering any kind of limit (peak, residual, etc.) thismodel is predictive. It has to be used only with levelvalues taken in the domain, e.g. 7oso21MPa.

6. Conclusions

Geostatistical analysis and simulation, taking intoaccount the directional anisotropy of the joint wallsurfaces morphology, were performed and cross vali-dated. Then the heights and derivatives variograms,before and after kriging, were compared. On this basis, amultidimensional statistical analysis permits to sort out,among the morphological factors, the more importantones. Only the factors that depend on shear directionwere kept to characterise the variations in joint shearbehaviour with changing shear direction.Geostatistical analysis permits to evaluate, in each

direction, the morphological factors. These factors areassociated to normal stress and shear displacement tobuild up an experimental design. The responses, shearstress and dilatancy, are chosen to represent the shearbehaviour.The principal result is a model (Morphological factor

Z4 Normal Stress Horizontal Displacement, and theirinteractions) in good agreement with the tests. In otherword, it is possible, with this model, to simulate theshear behaviour along the five classical phases. With thisexperimental design the mechanical part of the beha-viour, shear stress and dilatancy, is well simulated intaking into account one morphological and twomechanical factors (and their interactions). As a matterof fact, dilatancy is obtained with a better agreementthan shear stress. This is to be corrected with adding the‘‘zero points’’, i.e. shear stress equal to 0MPa when thehorizontal displacement is null, in the multilinearregression.Another important result is that the morphological

factor Z3 is not required in a model representing thejoint shear behaviour.The two morphological factors RL and y2 contain

only a part of the necessary information on themorphology: the height for RL and angularity for y2:According to the experimental design results, Z4 alreadyhas this information. These two factors are not requiredfor a model that contains the Z4 morphological factor.This morphology capability of Z4 is explained by its

computation that considers only the real area in contactof the fracture walls. Then, this model could be used topredict a Mohr–Coulomb envelope curve in the Mohrdiagram. The actual limitations are the extreme valuesof the factors, e.g. 7–21MPa for the normal stress.It would be necessary to add more tests, performed

for example in between 0 and 7MPa to be able tosimulate the shear behaviour within these values.

Acknowledgements

This work was supported by an NSERC grant toG. Archambault.

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S

F

+ −

Fig. 10. Morphological factor, Z4; capabilities.

0

5

10

15

20

25

0

σ (MPa)

τ (M

Pa)

7 MPa

21 MPa

10 20 30

Fig. 11. Simulated peak envelope curve (Mohr–Coulomb criterion).

P. Lopez et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 947–954 953

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