Mechanics and fluid transport in a degradable discontinuity

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1 Engineering Geology 53 (1999) Mechanics and fluid transport in a degradable discontinuity A.P.S. Selvadurai *, T.S. Nguyen 1 Department of Civil Engineering and Applied Mechanics, McGill University, 817 Sherbrooke Street West, Montreal, Que. H3A 2K6, Canada Abstract This paper briefly examines the mechanical and hydraulic response of discontinuity in a geologic medium that can experience degradation of its contacting regions due to shear and normal stresses acting on it. The paper discusses the methodologies that can be adopted to describe the mechanics and fluid transport behaviour of such a joint. The calibration capabilities of the model are illustrated by appeal to a comparison of experimental results with computational predictions Elsevier Science B.V. All rights reserved. Keywords: Elasto plastic stress strain relationship; Finite-element modelling; Gouge production; Hydraulic conductivity; Rock joint mechanics 1. Introduction increments of stress and the increment of strain. These relationships are derived from direct shear The study of joints and discontinuities in geo- tests under constant normal stress. The second logic media has a long history in terms of experi- category of constitutive relationships are the mental studies, field investigations, theoretical elasto plastic models derived by appeal to a theory developments and computational modelling (e.g. of plasticity. These models are characterized by see Stephansson, 1985; Barton and Stephansson, recoverable elastic behaviour of the joint prior to 1990; Selvadurai and Boulon, 1995). The studies sliding and irrecoverable plastic behaviour during by Patton (1966), Ladanyi and Archambault sliding. The plastic deformations are derived from (1970), and Barton and Choubey (1977) investi- a yield criterion and a flow rule applicable to an gated the mechanical response of joints subjected interface. The literature in geomechanics contains to a range of stress states that can induce dilatancy an extensive record of the applications of plasticityor degradation at the asperities of both conforming based models for the study of interface phenomena and non-conforming surfaces. The incremental (Selvadurai and Boulon, 1995). Among these relationships (e.g. see Goodman and Dubois, 1972) developments, the model proposed by Plesha involve piecewise linear relationships between the ( 1987) is particularly attractive owing to its simplicity and its ability to capture certain fundamen- * Corresponding author. Tel.: ; tal aspects of the mechanical behaviour of actual fax: joints, including dilatancy under shear and soften- address: apss@civil.lan.mcgill.ca (A.P.S. Selvadurai) ing due to asperity degradation. For developing 1 Permanent address: Wastes and Impacts Division, The Atomic Energy Control Board of Canada, 280 Slater Street, the hydraulic behaviour of rock joints, the parallel Ottawa, Ont. K1P 5S, Canada. plate model, developed by considerations of flow /99/$ see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S (99)00040-X

2 244 A.P.S. Selvadurai, T.S. Nguyen / Engineering Geology 53 (1999) of an incompressible Newtonian viscous fluid between two parallel plates, is employed. This procedure is widely used to calculate the effective permeability k of a fracture (e.g. see Gale, 1982; Benjelloun, 1991). The permeability of the joint is thus expressed as a function of its effective opening to fluid flow, termed the hydraulic aperture. Since a natural fracture is unlike the parallel plate idealization, the hydraulic aperture is different from the mechanical aperture. In this paper, we adopt the methodology proposed by Plesha (1987) to formulate the elasto plastic stress strain relationship for a rock joint. We illustrate a procedure whereby most of the parameters required to postulate the elasto plastic Fig. 1. The joint model. constitutive relationships for the joint could be estimated from two widely used empirical coefficients proposed by Barton and Choubey (1977) the saw tooth joint is and Bandis et al. (1981) (i.e. the JRC and JCS). In order to extend these developments to consideration of the influence of gouge production on (1) F= s sin a1+t cos a1 +tan w (t cos a1 s sin a1). hydraulic conductivity, we note that gouge production in the joint can be related to plastic work Similarly, the plastic potential is defined as of the shear forces. The extended version of the Q= s sin a1+t cos a1. (2) model proposed by Plesha (1987) is implemented in the finite element code FRACON (fracture When plastic deformations occur, the asperities of consolidation). The analysis is restricted to joints the joint are damaged and such degradation pro- without infilling. cesses can result in a decrease in the asperity angle. Plesha (1987) assumed that, during plastic deformations, the asperity angle varies according to the relationship 2. The elasto plastic behaviour of the rock joints a1=a1 0 exp A P 0 Wp c dwp B, (3) In this study a dilatant rock joint is assumed to where a1 is the original asperity angle, c is a 0 contain surface asperities that are irregular in degradation coefficient and Wp is the plastic work shape and height. A perfectly matching joint will of the shear stresses such that have surfaces that can be idealized by the twodwp=t dimensional saw tooth pattern proposed by Patton du, (4) (1966). The asperities are assumed to have regular where u is the relative shear displacement at the angles of inclination a1 with respect to the horizontal joint. direction ( Fig. 1). Along a typical asperity, When asperity damage is considered, strain inclined at an angle a1, the relationship between softening phenomena can occur and the yield the macro values of the stresses t and s the local function F and the plastic potential Q will depend or micro values of the stresses t and s can be on t, s and Wp such that a a obtained by appeal to local equilibrium at the F=F(t, s, Wp) Q=Q(t, s, Wp). (5) inclined sliding plane. Avoiding details of calculation, it can be shown that the yield criterion for Following conventional developments, it can be

3 A.P.S. Selvadurai, T.S. Nguyen / Engineering Geology 53 (1999) shown that the increments of elastic displacement are given by ds i =De ij due j, (6) shear stress t under constant normal stress peak conditions) proposed by Barton et al. (1985), it can be shown that where ds are the incremental stresses (ds =dt i 1 and ds =ds); and the increments of plastic dis- 2 placement are given by dup=dl Q, (7) i s i k s = t peak u peak = s tan CJRC log 10A JCS s B +w D L 500 A JRC L B 1/3 where where L is the size of the sample. The normal stiffness k can be determined by n dl= 1 F De du (8) performing uniaxial compression tests on both y H s ij j i intact and jointed specimens (Bandis et al., 1981). It can be shown that the normal stiffness at any y= F Q De ; H= F s lm s Wp t Q t. (9) level of normal stress is l m Consequently, we can write Eq. (9) in the form k = ds n dv nia =k 1 s v k m ni +sb 2. (14) ds =Dep du = i ij j ADe ij 1 Q F De De y H s il mj s l m B. The parameters k and v can be determined ni m from compression tests on jointed rock samples (10) (e.g. see Benjelloun, 1991; Boulon et al., 1993). There is limited experimental data on the determination of the degradation constant c. Experiments 3. Mechanical parameters for the joint model by Hutson and Dowding (1990) and Qiu et al. ( 1993) indicate that c increases with the normal The parameters governing the model proposed stress, whereas Benjelloun (1991) observed a by Plesha (1987) are the elastic stiffnesses, the reverse trend. It is likely that c will be related to degradation factor c, the initial asperity angle a1 JRC, JCS and s. However, further experimental and the friction angle w. Assuming symmetry contionship studies are necessary to establish clearly the rela- siderations of induced elastic behaviour during between c and s. imposed shear, it can be shown that De =k ; De =k ; De =De =0, ij s 22 n (11) 4. Hydraulic behaviour of the rock joint where k and k are elastic shear and normal s n stiffnesses. Plesha (1987) estimated these model The conventional parallel plate model develparameters by calibrating the results of experimenpressible oped by consideration of laminar flow of an incom- tal data derived from shear tests under constant Newtonian viscous fluid between two normal stress. It can be shown, if no asperity plates with parallel flat surfaces gives the relation- degradation occurs before peak shear stress is ship mobilized, that k= e2 h 12, (15) a1=jrc log 0 10A JCS s B. (12) where e is the hydraulic aperture. h Using the empirical relationship for u (the The empirical relationship proposed by Barton peak shear displacement corresponding to the peak (1982) relates the hydraulic aperture e to the h, (13)

4 246 A.P.S. Selvadurai, T.S. Nguyen / Engineering Geology 53 (1999) mechanical aperture e m according to 5. Computational modelling e = e2 m The basic concepts advocated for the mechan- h JRC5/2, (16) ical and hydraulic behaviour of the joint were implemented in the finite element code FRACON where e and e are expressed in micrometres. h m (Selvadurai and Nguyen, 1995; Nguyen, 1995) The [Note that Eq. (16) is an empirical relationship; as code is a finite element scheme for the computasuch, the apparent dimensional inconsistency tional modelling of coupled thermal, hydraulic and should be disregarded.] mechanical (THM ) behaviour of both intact and Witherspoon et al. (1979) proposed a linear jointed rock masses. The THM processes in the relationship between the hydraulic and mechanical intact geomaterial are represented by an extension apertures of the form to the generalized theory of poroelasticity proposed by Biot (1941) (see also Selvadurai, 1996) h h0 m e =e +fde, (17) and the flow in fractures is by consideration of a where e is the initial hydraulic aperture, De is h0 m parallel plate simulation. There is full thermo the variation in the mechanical aperture due to the hydro-mechanical coupling between the fractures combined effects of compression and shear and f is and the intact poroelastic geomaterial. In addition, a constant. Benjelloun (1991) has confirmed the the fractures exhibit non-linear constraints through validity of Eq. (17) and found that f varies between the interface plasticity simulations described pre- 0.5 and 1.0. The factor takes into consideration the viously. Such variable non-linear interface consurface regularity of the parallel plates. The factor straints also account for the variations in the fluid f=1 applies to the case of parallel plates with no transport characteristics at the joints. surface irregularities (i.e. relatively open joint with The influence of joint shearing on the permeabilapertures of the order of millimetres). For most ity characteristics of rock joints composed of gneiss other cases, f<1 and depends on the geometry of was investigated by Bandis et al. (1985), using a the flow path. For rectilinear flow f generally uniaxial cell. A schematic configuration of the cell approaches 0.8 and for radial flow f approaches 0.5. In this paper, we propose a relationship that is intended to account for (i) variation in joint permeability due to the dilation of the joint, which should increase with joint shear, and (ii) decrease in the joint permeability, which results in asperity degradation and gouge production. This latter aspect is considered to be a novel development and intended to model observations of the hydraulic behaviour of joints made by Bandis et al. (1985). The basic theoretical justification rests on the assumption that gouge production is related to the plastic work of shear tractions on the joint. We assume that f in Eq. (17) can be expressed in the form f=f 0 exp A P 0 Wp cf dwp B, (18) where c is a gouge production factor. There is a f high likelihood that the additional parameters f 0 and c may also be related to JRC, JCS and s. f This assertion can only be verified through an extensive program of experimental research. Fig. 2. Schematic configuration of the experimental facility used to perform hydromechanical experiments on gneiss (Bandis et al., 1985; Makurat et al., 1990).

5 A.P.S. Selvadurai, T.S. Nguyen / Engineering Geology 53 (1999) Fig. 3. Finite element model for joint shear under constant normal stress. is shown in Fig. 2. The joint sample was first compressed by equal biaxial normal stresses. The joint was then sheared by maintaining one load constant and increasing the other. At specific values of the shear displacement, the permeability of the joint was determined by injecting water through the joint and measuring the flow rate. Bandis et al. (1985) recorded the evolution of the joint permeability with increasing shear displacement. Although both the normal and shear stresses varied during the experiment, Bandis et al. (1985) assumed constant normal stress conditions to sim- ulate the evolution of joint permeability during one typical experiment. The assumed constant normal stress was the average value of the actual stress. In this study, we computationally simulate the experiment by assuming constant normal stress conditions. The finite element model used in the FRACON simulation is shown in Fig. 3. The joint properties estimated by Bandis et al. (1985) are as follows: JRC=7; JCS=110 MPa, joint length L= 0.15 m, average normal stress 1.5 MPa. In addition to these parameters, an asperity degradation factor c= m/n is assumed. The joint exhibits dilatancy during shear. The dilation calcu- lated by FRACON is shown in Fig. 4. This dilation is accompanied by a corresponding increase in the permeability of the joint ( Fig. 5). However, this permeability decreases with continued shear defor- mation, as gouge development takes place by asperity breakage. In their studies, Bandis et al. ( 1985) could not simulate this permeability decrease, purely by appeal to the model proposed by Barton (1982). The FRACON code simulation, with no gouge production (c =0) produces results f similar to those presented by Bandis et al. (1985). By assuming f =1 and c =0.001 m/n [see Eq. 0 f (18)], the FRACON code is able to provide predictions, where the trends agree well with the experimental data. Most importantly, the tendency for the gouge-induced reduction in the permeability of the joint with increasing relative shear u is correctly predicted. Further details can be found in the article by Nguyen and Selvadurai (1998). 6. Concluding remarks Fig. 4. Shear dilation calculated via the FRACON code for the experiment conducted by Bandis et al. (1985). The concept of a degradable discontinuity attempts to examine the influence of asperity

6 248 A.P.S. Selvadurai, T.S. Nguyen / Engineering Geology 53 (1999) are not intended to be validations of the model; instead, the trends observed in the computational simulations address several aspects of the alteration of the permeability characteristics of jointinduced asperity degradation and gouge production. The computational simulations adequately address the behaviour of the joint under constant normal stress and constant normal stiffness. Such implementations are also able to address the influence of shear on joint permeability and scale effects. The methodology provides a useful extension to the current state of characterization of shear-induced irreversible alterations in the permeability of a joint. Fig. 5. Effect of shear on the joint permeability: FRACON code simulation of the experiment by Bandis et al. (1985). Acknowledgements The work was carried out with the joint support of the Atomic Energy Control Board of Canada and The Natural Sciences and Engineering Research Council of Canada (NSERC), as a part of the ongoing research programme into Thermal Consolidation Around Nuclear Waste Repositories. The views expressed in the paper are those of the authors and do not in any way reflect the views or policies of the Atomic Energy Control Board of Canada or NSERC. degradation on the hydraulic behaviour of a joint. The paper reviews the formulation of the mechanics of a joint by recourse to an incremental theory of plasticity applicable to a dilatant joint where asperity degradation is coupled to the plastic work of shear forces. The basic methodology is also extended to cover the alterations in the hydraulic References conductivity of the joint by initial dilation of the joint due to shear and the subsequent production Bandis, S., Lumsen, A.C., Barton, N., Experimental of gouge material during asperity degradation. The studies of scale effects on the shear behaviour of rock joints. paper proposes a postulate whereby the hydraulic Int. J. Rock Mech. Mining Sci. Geomech. Abstr. 18, Bandis, S., Makurat, A., Vik, G., Predicted and measured aperture varies with the plastic work of shear hydraulic conductivity of rock joints. In: Stephansson, O. forces. Two additional material parameters are (Ed.), Proceedings of the International Symposium on required to implement the characteristics related Fundamentals of Rock Joints, Bjorkliden, Sweden. A.A. to gouge-production-induced alteration in the per- Balkema, pp meability of the joint. The parameters governing Barton, N., Modelling rock joint behaviour from in situ block tests; implications for nuclear waste repository design. the mechanical response of the joint are defined in Office of Nuclear Waste Isolation Report, ONWI-308. relation to the joint friction angle w, the empirical Barton, N., Choubey, V., The shear strength of rock joints measures JRC and JCS and their variations. The in theory and practice. Rock Mech. 10, constitutive model for the degradable discontinuity Barton, N., Bandis, S., Bakhtar, K., Strength deformation is implemented in the two-dimensional finite ele- and conductivity of rock joints. Int. J. Rock Mech. Mining Sci. Geomech. Abstr. 22, ment code FRACON. The performance of the Barton, N., Stephansson, O. (Eds.), Rock Joints; Proceedings model is calibrated against a set of experimental International Symposium on Rock Joints, Norway results reported in the literature. These correlations A.A. Balkema, The Netherlands.

7 A.P.S. Selvadurai, T.S. Nguyen / Engineering Geology 53 (1999) Benjelloun, Z.H., Etude experimentale et modéllisation hydrological mechanical processes in geological media, du comportement hydromécanique des joints rocheux. PhD Thesis, McGill University. Thèse de doctorat, Université Joseph Fourier, Grenoble I. Nguyen, T.S., Selvadurai, A.P.S., A model for coupled Biot, M.A., General theory of three dimensional consoli- mechanical and hydraulic behaviour of a rock joint. Int. J. dation. J. Appl. Phys. 12, Numer. Anal. Methods Geomech. 22, Boulon, M.J., Selvadurai, A.P.S., Benjelloun, H., Feuga, B., Patton, F.D., Multiple modes of shear failure in rock, Influence of rock joint degradation on hydraulic con- Proceedings of the 1st Congress of International Society of ductivity. Int. J. Rock Mech. Mining Sci. Geomech. Abstr. Rock Mechanics, Lisbon vol. 1., , Plesha, M.E., Constitutive models for rock discontinuities Gale, J.E., Assessing the permeability characteristics of with dilatancy and surface degradation. Int. J. Numer. Anal. fractured rock, Recent Trends in Hydrogeology. The Geolo- Methods Geomech. 11, gical Society of America, Boulder, CO, pp Special Qiu, X., Plesha, M.E., Haimson, B.C., Huang, X., An paper 189 investigation of the mechanics of rock joints part II, Goodman, R.E., Dubois, J., Duplication of dilatancy in analytic investigation. Int. J. Rock Mech. Mining Sci Geomech. Abstr. 30, analysis of jointed rock. J. Soil Mech. Fdn. Div. Proc. ASCE Selvadurai, A.P.S. (Ed.), Mechanics of Poroelastic Media SM4, Kluwer, The Netherlands. Hutson, R.W., Dowding, C.H., Joint asperity degradation Selvadurai, A.P.S., Boulon, M.J. (Eds.), Mechanics of Geomatduring cyclic shear. Int. J. Rock Mech. Mining Sci. erial Interfaces, Studies in Applied Mechanics vol. 42, 1995, Geomech. Abstr. 27, Elsevier, Amsterdam, The Netherlands. Ladanyi, B., Archambault, G., Simulation of shear behav- Selvadurai, A.P.S., Nguyen, T.S., Computational modeliour of a jointed rock mass, Proc. 11th Symposium on Rock ling of isothermal consolidation of fractured porous media. Mechanics. American Institute of Mining Engineers, New Comput. Geotechnics 17, York, pp Stephansson, O. (Ed.), Proceedings, International Symposium Makurat, A., Barton, N., Rad, N.S., Bandis, S., Joint on Fundamentals of Rock Joints, Bjerkliden, Sweden conductivity variation due to normal and shear deforma- A.A. Balkema, The Netherlands. tion. In: Barton, N., Stephansson, O. (Eds.), Proceedings Witherspoon, P.A., Wang, J.S.V., Iwai, K., Gale, J.E., of the International Symposium on Rock Joints, Loen, Validity of cubic law for potential size effects in experimental Norway. A.A. Balkema, pp determination of the hydraulic properties of a fracture. Nguyen, T.S., Computational modelling of thermal Water Resources Res.

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